New constructions of nonregular cospectral graphs

Suleiman Hamud; Abraham Berman

doi:10.1515/spma-2023-0109

Article Open Access

New constructions of nonregular cospectral graphs

Suleiman Hamud and Abraham Berman

Published/Copyright: February 22, 2024

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Special Matrices Volume 12 Issue 1

Abstract

We consider two types of joins of graphs G 1 and G 2 , G 1 ⊻ G 2 – the neighbors splitting join and G 1 ∨ = G 2 – the nonneighbors splitting join, and compute the adjacency characteristic polynomial, the Laplacian characteristic polynomial, and the signless Laplacian characteristic polynomial of these joins. When G 1 and G 2 are regular, we compute the adjacency spectrum, the Laplacian spectrum, the signless Laplacian spectrum of G 1 ∨ = G 2 , and the normalized Laplacian spectrum of G 1 ⊻ G 2 and G 1 ∨ = G 2 . We use these results to construct nonregular, nonisomorphic graphs that are cospectral with respect to the four matrices: adjacency, Laplacian, signless Laplacian and normalized Laplacian.

Keywords: spectra of graphs; nonisomorphic cospectral graphs; NS and NNS joins; coronal and Schur complement

MSC 2010: 05C50

1 Introduction

Spectral graph theory is the study of graphs via the spectrum of matrices associated with them [3,6,8,22,27]. The graphs in this article are undirected and simple. There are several matrices associated with a graph, four of which are considered here: the adjacency matrix, the Laplacian matrix, the signless Laplacian matrix, and the normalized Laplacian matrix.

Let G = ( V ( G ) , E ( G ) ) be a graph with vertex set V ( G ) = { v 1 , v 2 , … , v n } and edge set E ( G ) .

Definition 1.1

The adjacency matrix of G , A ( G ) , is defined as follows:

( A ( G ) ) i j = 1 , if v i and v j are adjacent ; 0 , otherwise.

Let d i = d G ( v i ) be the degree of vertex v i in G , and let D ( G ) be the diagonal matrix with diagonal entries d 1 , d 2 , … , d n .

Definition 1.2

The Laplacian matrix, L ( G ) , and the signless Laplacian matrix, Q ( G ) , of G are defined as L ( G ) = D ( G ) − A ( G ) and Q ( G ) = D ( G ) + A ( G ) , respectively.

Definition 1.3

[6] The normalized Laplacian matrix, ℒ ( G ) , is defined to be ℒ ( G ) = I n − D ( G ) − 1 2 A ( G ) D ( G ) − 1 2 (with the convention that if the degree of vertex v i in G is 0, then ( d i ) − 1 2 = 0 ). In other words,

( ℒ ( G ) ) i j = 1 , if i = j and d i ≠ 0 ; − 1 d i d j , if i ≠ j and v i is adjacent to v j ; 0 , otherwise .

Notation 1.4

For an n × n matrix M , we denote the characteristic polynomial det ( x I n − M ) of M by f M ( x ) , where I n is the identity matrix of order n . In particular, for a graph G , f X ( G ) ( x ) is the X -characteristic polynomial of G , for X ∈ { A , L , Q , ℒ } . The roots of the X -characteristic polynomial of G are the X -eigenvalues of G and the collection of the X -eigenvalues, including multiplicities, is called the X -spectrum of G .

Notation 1.5

The multiplicity of an eigenvalue λ is denoted by a superscript above λ .

Example 1.6

The A -spectrum of the complete graph K n is { n − 1 , ( − 1 ) [ n − 1 ] } .

Remark 1.7

λ 1 ( G ) ≥ λ 2 ( G ) ≥ ⋯ ≥ λ n ( G ) ,

μ 1 ( G ) ≤ μ 2 ( G ) ≤ ⋯ ≤ μ n ( G ) ,

ν 1 ( G ) ≥ ν 2 ( G ) ≥ ⋯ ≥ ν n ( G ) ,

δ 1 ( G ) ≤ δ 2 ( G ) ≤ ⋯ ≤ δ n ( G )

are the eigenvalues of A ( G ) , L ( G ) , Q ( G ) , and ℒ ( G ) , respectively. Then, ∑ i = 1 n λ i = 0 , μ 1 ( G ) = 0 , ν n ( G ) ≥ 0 , and δ 1 ( G ) = 0 , δ n ( G ) ≤ 2 (equality iff G is bipartite).

Remark 1.8

If G is an r-regular graph, then μ i ( G ) = r − λ i ( G ) , ν i ( G ) = r + λ i ( G ) , and δ i ( G ) = 1 − 1 r λ ( G ) .

Definition 1.9

Two graphs G and H are X -cospectral if they have the same X -spectrum . If X -cospectral graphs are not isomorphic, we say that they are XNICS.

Definition 1.10

Let S be a subset of { A , L , Q , ℒ } . The graphs G and H are SNICS if they are XNICS for all X ∈ S .

Definition 1.11

A graph G is determined by its X -spectrum if every graph H that is X -cospectral with G is isomorphic to G .

A basic problem in spectral graph theory, [28, 29], is determining which graphs are determined by their spectrum or finding nonisomorphic X -cospectral graphs.

Theorem 1.12

[28] If G is regular, then the following are equivalent;

G is determined by its A-spectrum,
G is determined by its L-spectrum,
G is determined by its Q-spectrum,
G is determined by its ℒ -spectrum.

Thus, for regular graphs G and H , we say that G and H are cospectral if they are X -cospectral with respect to any X ∈ { A , L , Q , ℒ } .

Proposition 1.13

[28] Every regular graph with less than ten vertices is determined by its spectrum.

Example 1.14

Graphs in Figure 1 are regular and cospectral. They are nonisomorphic since in G , there is an edge that lies in three triangles but there is no such edge in H .

Figure 1

Two regular nonisomorphic cospectral graphs. (a) ( G ) and (b) ( H ) .

In recent years, several researchers studied the spectral properties of graphs which are constructed by graph operations. These operations include disjoint union, the Cartesian product, the Kronecker product, the strong product, the lexicographic product, the rooted product, the corona, the edge corona, the neighborhood corona, etc. We refer the reader to [1,2,8,9,12,13,16,21,23–26] and the references therein for more graph operations and the results on the spectra of these graphs.

Many operations are based on the join of graphs.

Definition 1.15

[14] The join of two graphs is their disjoint union together with all the edges that connect all the vertices of the first graph with all the vertices of the second graph.

Recently, many researchers provided several variants of join operations of graphs and investigated their spectral properties. Some examples are Cardoso [5], Indulal [17], Liu and Zhang [18], Varghese and Susha [30], and Das and Panigrahi [11].

Butler [4] constructed nonregular bipartite graphs, which are cospectral with respect to both the adjacency and the normalized Laplacian matrices. He asked for examples of nonregular { A , L , ℒ } NICS graphs. A slightly more general question is

Question 1.16

Construct nonregular { A , L , Q , ℒ } NICS graphs.

Such examples can be constructed using special join operation defined by Lu et al. [19] and a variant of this operation, suggested in this article.

Definition 1.17

[19] Let G 1 and G 2 be two vertex disjoint graphs with V ( G 1 ) = { u 1 , u 2 , … , u n } . The splitting V -vertex join of G 1 and G 2 , denoted by G 1 ⊻ G , is obtained by adding new vertices u 1 ′ , u 2 ′ , … , u n ′ to G 1 ∨ G 2 and connecting u i ′ to u j if and only if ( u i , u j ) ∈ E ( G 1 ) .

We refer to the splitting V -vertex join as neighbors splitting (NS) join and define a new type of join, nonneighbors splitting (NNS) join.

Definition 1.18

Let G 1 and G 2 be two vertex disjoint graphs with V ( G 1 ) = { u 1 , u 2 , … , u n } . The NNS join of G 1 and G 2 , denoted by G 1 ∨ = G 2 , is obtained by adding new vertices u 1 ′ , u 2 ′ , … , u n ′ to G 1 ∨ G 2 and connecting u i ′ to u j iff ( u i , u j ) ∈ E ( G 1 ) .

Example 1.19

Let G 1 and G 2 be the path P 4 and the path P 2 , respectively. The graphs P 4 ∨ = P 2 and P 4 ⊻ P 2 are given in Figure 2.

$Figure 2 The graphs P 4 ⊻ P 2 {P}_{4}\hspace{0.33em}⊻\hspace{0.33em}{P}_{2} and P 4 ∨ = P 2 {P}_{4}\mathop{\vee }\limits_{=}{P}_{2} . (a) P 4 ⊻ P 2 {P}_{4}\hspace{0.33em}⊻\hspace{0.33em}{P}_{2} and (b) P 4 ∨ = P 2 {P}_{4}\mathop{\vee }\limits_{=}{P}_{2} .$

Figure 2

The graphs P 4 ⊻ P 2 and P 4 ∨ = P 2 . (a) P 4 ⊻ P 2 and (b) P 4 ∨ = P 2 .

The structure of the article is as follows; after preliminaries, we compute the adjacency characteristic polynomial, the Laplacian characteristic polynomial, and the signless Laplacian characteristic polynomial of G 1 ∨ = G 2 and G 1 ⊻ G 2 , and use the results to construct { A , L , Q } NICS graphs, and finally, under regularity assumptions, we compute the A -spectrum, the L -spectrum, the Q -spectrum, and the ℒ -spectrum of NS and NNS joins and use the results to construct { A , L , Q , ℒ } NICS graphs.

2 Preliminaries

Notation 2.1

1 n denotes n × 1 column whose all entries are 1,
J s × t = 1 s 1 t T , J s = J s × s ,
O s × t denotes the zero matrix of order s × t ,
adj ( A ) denotes the adjugate of A .
G ¯ denotes the complement of graph G.

Definition 2.2

[7,20] The coronal Γ M ( x ) of an n × n matrix M is the sum of the entries of the inverse of the characteristic matrix of M , i.e.,

(2.1) Γ M ( x ) = 1 n T ( x I n − M ) − 1 1 n .

Lemma 2.3

[7,20] Let M be an n × n matrix with all row sums equal to r (e.g., the adjacency matrix of a r-regular graph). Then,

Γ M ( x ) = n x − r .

Definition 2.4

Let M be a block matrix

M = A B C D

such that its blocks A and D are square. If A is invertible, the Schur complement of A in M is

M ∕ A = D − C A − 1 B ,

and if D is invertible, the Schur complement of D in M is

M ∕ D = A − B D − 1 C .

Issai Schur proved the following lemma.

Lemma 2.5

[15] If D is invertible, then

det M = det ( M ∕ D ) det D ,

and if A is invertible, then

det M = det ( M ∕ A ) det A .

Lemma 2.6

Let M be a block matrix

M = A B J n 1 × n 2 B C O n 1 × n 2 J n 2 × n 1 O n 2 × n 1 D ,

where A, B, and C are square matrices of order n 1 and D is a square matrix of order n 2 . Then, the Schur complement of x I n 2 − D in the characteristic matrix of M is

x I n 1 − A − Γ D ( x ) J n 1 − B − B x I n 1 − C .

Proof

The characteristic matrix of M is

x I 2 n 1 + n 2 − M = x I n 1 − A − B − J n 1 × n 2 − B x I n 1 − C O n 1 × n 2 − J n 2 × n 1 O n 2 × n 1 x I n 2 − D .

The Schur complement of ( x I n 2 − D ) is

□ ( x I 2 n 1 + n 2 − M ) / ( x I n 2 − D ) = x I n 1 − A − B − B x I n 1 − C − − J n 1 × n 2 O n 1 × n 2 ( x I n 2 − D ) − 1 − J n 2 × n 1 O n 2 × n 1 = x I n 1 − A − B − B x I n 1 − C − 1 n 1 1 n 2 T O n 1 × n 1 ( ( x I n 2 − D ) − 1 ) 1 n 2 1 n 1 T O n 1 × n 1 = x I n 1 − A − Γ D ( x ) J n 1 − B − B x I n 1 − C .

Lemma 2.7

[8] If A is an n × n real matrix and a is an real number, then

(2.2) det ( A + α J n ) = det ( A ) + α 1 n T adj ( A ) 1 n .

3 The characteristic polynomials of the NNS and NS joins

Lu et al. [19] computed the adjacency, Laplacian, and signless Laplacian characteristic polynomials of G 1 ⊻ G 2 , where G 1 and G 2 are regular.

Here, we compute the characteristic polynomials of G 1 ∨ = G 2 and G 1 ⊻ G 2 , where G 1 and G 2 are arbitrary graphs. The proofs for the two joins (NS and NNS) are quite similar and use Lemma 2.5 (twice) and Lemmas 2.6 and 2.7. The results are used to construct nonregular { A , L , Q } NICS graphs.

3.1 Adjacency characteristic polynomial

Theorem 3.1

Let G i be a graph on n i vertices for i = 1 , 2 . Then,

f A ( G 1 ∨ = G 2 ) ( x ) = x n 1 f A ( G 2 ) ( x ) det x I n 1 − A ( G 1 ) − 1 x A 2 ( G ¯ 1 ) 1 − Γ A ( G 2 ) ( x ) Γ A ( G 1 ) + 1 x A 2 ( G ¯ 1 ) ( x ) .
f A ( G 1 ⊻ G 2 ) ( x ) = x n 1 f A ( G 2 ) ( x ) det x I n 1 − A ( G 1 ) − 1 x A 2 ( G 1 ) 1 − Γ A ( G 2 ) ( x ) Γ A ( G 1 ) + 1 x A 2 ( G 1 ) ( x ) .

Proof

We prove (a). The proof of (b) is similar.

With a suitable ordering of the vertices of G 1 ∨ = G 2 , we obtain

A ( G 1 ∨ = G 2 ) = A ( G 1 ) A ( G ¯ 1 ) J n 1 × n 2 A ( G ¯ 1 ) O n 1 × n 1 O n 1 × n 2 J n 2 × n 1 O n 2 × n 1 A ( G 2 ) .

Thus,

f A ( G 1 ∨ = G 2 ) ( x ) = det ( x I 2 n 1 + n 2 − A ( G 1 ∨ = G 2 ) ) = det x I n 1 − A ( G 1 ) − A ( G ¯ 1 ) − J n 1 × n 2 − A ( G ¯ 1 ) x I n 1 O n 1 × n 2 − J n 2 × n 1 O n 2 × n 1 x I n 2 − A ( G 2 ) = det ( x I n 2 − A ( G 2 ) ) det ( ( x I 2 n 1 + n 2 − A ( G 1 ∨ = G 2 ) ) / ( x I n 2 − A ( G 2 ) ) )

by the Lemma of Schur (Lemma 2.5).

By Lemma 2.6,

( x I 2 n 1 + n 2 − A ( G 1 ∨ = G 2 ) ) / ( x I n 2 − A ( G 2 ) ) = x I n 1 − A ( G 1 ) − Γ A ( G 2 ) ( x ) J n 1 × n 1 − A ( G ¯ 1 ) − A ( G ¯ 1 ) x I n 1 .

Using again Lemma 2.5, we obtain

det ( ( x I 2 n 1 + n 2 − A ( G 1 ∨ = G 2 ) ) / ( x I n 2 − A ( G 2 ) ) ) = det ( x I n 1 ) det x I n 1 − A ( G 1 ) − Γ A ( G 2 ) ( x ) J n 1 × n 1 − 1 x A 2 ( G ¯ 1 ) .

By Lemma 2.7, we obtain

det ( ( x I 2 n 1 + n 2 − A ( G 1 ∨ = G 2 ) ) / ( x I n 2 − A ( G 2 ) ) ) = x n 1 det ( x I n 1 − A ( G 1 ) − 1 x A 2 ( G ¯ 1 ) ) − Γ A ( G 2 ) ( x ) 1 n 1 T adj x I n 1 − A ( G 1 ) − 1 x A 2 ( G ¯ 1 ) 1 n 1 = x n 1 det x I n 1 − A ( G 1 ) − 1 x A 2 ( G ¯ 1 ) 1 − Γ A ( G 2 ) ( x ) 1 n 1 T ( x I n 1 − A ( G 1 ) − 1 x A 2 ( G ¯ 1 ) ) − 1 1 n 1 = x n 1 det x I n 1 − A ( G 1 ) − 1 x A 2 ( G ¯ 1 ) 1 − Γ A ( G 2 ) ( x ) Γ A ( G 1 ) + 1 x A 2 ( G ¯ 1 ) ( x ) .

Thus,

□ f A ( G 1 ∨ = G 2 ) ( x ) = x n 1 f A ( G 2 ) ( x ) det x I n 1 − A ( G 1 ) − 1 x A 2 ( G ¯ 1 ) 1 − Γ A ( G 2 ) ( x ) Γ A ( G 1 ) + 1 x A 2 ( G ¯ 1 ) ( x ) .

3.2 Laplacian characteristic polynomial

In this section, we derive the Laplacian characteristic polynomials of G 1 ⊻ G 2 and G 1 ∨ = G 2 when G 1 and G 2 are arbitrary graphs.

Theorem 3.2

Let G i be a graph on n i vertices for i = 1 , 2 . Then,

f L ( G 1 ∨ = G 2 ) ( x ) = det ( ( x − n 1 ) I n 2 − L ( G 2 ) ) det ( ( x − n 1 + 1 ) I n 1 + D ( G 1 ) ) ⋅ det ( ( x − n 1 − n 2 + 1 ) I n 1 − L ( G 1 ) + D ( G 1 ) − A ( G ¯ 1 ) ( ( x − n 1 + 1 ) I n 1 + D ( G 1 ) ) − 1 A ( G ¯ 1 ) ) ⋅ [ 1 − Γ L ( G 2 ) ( x − n 1 ) Γ L ( G 1 ) − D ( G 1 ) + A ( G ¯ 1 ) ( ( x − n 1 + 1 ) I n 1 + D ( G 1 ) ) − 1 A ( G ¯ 1 ) ( x − n 1 − n 2 + 1 ) ] .
f L ( G 1 ⊻ G 2 ) = det ( ( x − n 1 ) I n 2 − L ( G 2 ) ) det ( x I n 1 − D ( G 1 ) ) ⋅ det ( ( x − n 2 ) I n 1 − L ( G 1 ) − D ( G 1 ) − A ( G 1 ) ( x I n 1 − D ( G 1 ) ) − 1 A ( G 1 ) ) ⋅ [ 1 − Γ L ( G 2 ) ( x − n 1 ) Γ L ( G 1 ) + D ( G 1 ) + A ( G 1 ) ( x I n 1 − D ( G 1 ) ) − 1 A ( G 1 ) ( x − n 2 ) ] .

Proof

(a) With a suitable ordering of the vertices of G 1 ∨ = G 2 , we obtain

L ( G 1 ∨ = G 2 ) = ( n 1 + n 2 − 1 ) I n 1 + L ( G 1 ) − D ( G 1 ) − A ( G ¯ 1 ) − J n 1 × n 2 − A ( G ¯ 1 ) ( n 1 − 1 ) I n 1 − D ( G 1 ) O n 1 × n 2 − J n 2 × n 1 O n 2 × n 1 n 1 I n 2 + L ( G 2 ) .

The Laplacian characteristic polynomial is

f L ( G 1 ∨ = G 2 ) ( x ) = det ( x I 2 n 1 + n 2 − L ( G 1 ∨ = G 2 ) ) = det ( ( x − n 1 ) I n 2 − L ( G 2 ) ) det ( ( x I 2 n 1 + n 2 − L ( G 1 ∨ = G 2 ) ) / ( ( x − n 1 ) I n 2 − L ( G 2 ) ) )

by the Lemma of Schur (Lemma 2.5).

By Lemma 2.6,

( x I 2 n 1 + n 2 − L ( G 1 ∨ = G 2 ) ) / ( ( x − n 1 ) I n 2 − L ( G 2 ) ) = ( x − n 1 − n 2 + 1 ) I n 1 − L ( G 1 ) + D ( G 1 ) − Γ L ( G 2 ) ( x − n 1 ) J n 1 × n 1 A ( G ¯ 1 ) A ( G ¯ 1 ) ( x − n 1 + 1 ) I n 1 + D ( G 1 ) .

Using again Lemma 2.5, we obtain

det ( ( x I 2 n 1 + n 2 − L ( G 1 ∨ = G 2 ) ) / ( ( x − n 1 ) I n 2 − L ( G 2 ) ) ) = det ( ( x − n 1 + 1 ) I n 1 + D ( G 1 ) ) det ( B − Γ L ( G 2 ) ( x − n 1 ) J n 1 × n 1 ) ,

where

B = ( x − n 1 − n 2 + 1 ) I n 1 − L ( G 1 ) + D ( G 1 ) − A ( G ¯ 1 ) ( ( x − n 1 + 1 ) I n 1 + D ( G 1 ) ) − 1 A ( G ¯ 1 ) .

By Lemma 2.7, we obtain

det ( ( x I 2 n 1 + n 2 − L ( G 1 ∨ = G 2 ) ) / ( ( x − n 1 ) I n 2 − L ( G 2 ) ) ) = det ( ( x − n 1 + 1 ) I n 1 + D ( G 1 ) ) ( det ( B ) − Γ L ( G 2 ) ( x − n 1 ) 1 n 1 T adj ( B ) 1 n 1 ) = det ( ( x − n 1 + 1 ) I n 1 + D ( G 1 ) ) det ( B ) [ 1 − Γ L ( G 2 ) ( x − n 1 ) 1 n 1 T B − 1 1 n 1 ] = det ( ( x − n 1 + 1 ) I n 1 + D ( G 1 ) ) ⋅ det ( ( x − n 1 − n 2 + 1 ) I n 1 − L ( G 1 ) + D ( G 1 ) − A ( G ¯ 1 ) ( ( x − n 1 + 1 ) I n 1 + D ( G 1 ) ) − 1 A ( G ¯ 1 ) ) ⋅ [ 1 − Γ L ( G 2 ) ( x − n 1 ) Γ L ( G 1 ) − D ( G 1 ) + A ( G ¯ 1 ) ( ( x − n 1 + 1 ) I n 1 + D ( G 1 ) ) − 1 A ( G ¯ 1 ) ( x − n 1 − n 2 + 1 ) ] .

Thus,

f L ( G 1 ∨ = G 2 ) ( x ) = det ( ( x − n 1 ) I n 2 − L ( G 2 ) ) det ( ( x − n 1 + 1 ) I n 1 + D ( G 1 ) ) ⋅ det ( ( x − n 1 − n 2 + 1 ) I n 1 − L ( G 1 ) + D ( G 1 ) − A ( G ¯ 1 ) ( ( x − n 1 + 1 ) I n 1 + D ( G 1 ) ) − 1 A ( G ¯ 1 ) ) ⋅ [ 1 − Γ L ( G 2 ) ( x − n 1 ) Γ L ( G 1 ) − D ( G 1 ) + A ( G ¯ 1 ) ( ( x − n 1 + 1 ) I n 1 + D ( G 1 ) ) − 1 A ( G ¯ 1 ) ( x − n 1 − n 2 + 1 ) ] .

The proof of (b) is similar.□

3.3 Signless Laplacian characteristic polynomial

Theorem 3.3

Let G i be a graph on n i vertices for i = 1 , 2 . Then,

f Q ( G 1 ∨ = G 2 ) ( x ) = det ( ( x − n 1 ) I n 2 − Q ( G 2 ) ) det ( ( x − n 1 + 1 ) I n 1 + D ( G 1 ) ) ⋅ det ( ( x − n 1 − n 2 + 1 ) I n 1 − Q ( G 1 ) + D ( G 1 ) − A ( G ¯ 1 ) ( ( x − n 1 + 1 ) I n 1 + D ( G 1 ) ) − 1 A ( G ¯ 1 ) ) ⋅ [ 1 − Γ Q ( G 2 ) ( x − n 1 ) Γ Q ( G 1 ) − D ( G 1 ) + A ( G ¯ 1 ) ( ( x − n 1 + 1 ) I n 1 + D ( G 1 ) ) − 1 A ( G ¯ 1 ) ( x − n 1 − n 2 + 1 ) ] .
f Q ( G 1 ⊻ G 2 ) = det ( ( x − n 1 ) I n 2 − Q ( G 2 ) ) det ( x I n 1 − D ( G 1 ) ) ⋅ det ( ( x − n 2 ) I n 1 − Q ( G 1 ) − D ( G 1 ) − A ( G 1 ) ( x I n 1 − D ( G 1 ) ) − 1 A ( G 1 ) ) ⋅ [ 1 − Γ Q ( G 2 ) ( x − n 1 ) Γ Q ( G 1 ) + D ( G 1 ) + A ( G 1 ) ( x I n 1 − D ( G 1 ) ) − 1 A ( G 1 ) ( x − n 2 ) ] .

Proof

The proof is similar to the proof of Theorem 3.2.□

Corollary 3.4

Let F and H be r-regular nonisomorphic cospectral graphs. Then, for every G,

G ⊻ F and G ⊻ H are { A , Q , L } NICS.
G ∨ = F and G ∨ = H are { A , Q , L } NICS.

Proof

(a) G ⊻ F and G ⊻ H are nonisomorphic since F and H are nonisomorphic. By Theorems 3.1, 3.2, and 3.3, f A ( G ⊻ F ) ( x ) = f A ( G ⊻ H ) ( x ) , f L ( G ⊻ F ) ( x ) = f L ( G ⊻ H ) ( x ) , and f Q ( G ⊻ F ) ( x ) = f Q ( G ⊻ H ) ( x ) since the matrices A ( F ) and A ( H ) have the same coronal (Lemma 2.3) and the same characteristic polynomial. This completes the proof of (a).

The proof of (b) is similar.□

Corollary 3.5

Let F and H be r-regular nonisomorphic cospectral graphs. Then, for every G,

F ⊻ G and H ⊻ G are { A , Q , L } NICS.
F ∨ = G and H ∨ = G are { A , Q , L } NICS.

Proof

The proof is similar to the proof of Corollary 3.4.□

Remark 3.6

The following examples demonstrate the importance of the regularity of the graphs F and H .

Example 3.7

The graphs F and H in Figure 3 are nonregular and A -cospectral [10]. The joins K 2 ⊻ F and K 2 ⊻ H in Figure 4 are not A -cospectral since the A -spectrum of K 2 ⊻ F is { − 2.2332 , − 2 , − 1.618 , 0 [ 3 ] , 0.577 , 0.618 , 4.6562 } and the A -spectrum of K 2 ⊻ H is { − 2.7039 , − 1.618 , − 1.2467 , 0 [ 3 ] , 0.2526 , 0.618 , 4.698 } . The joins K 2 ∨ = F and K 2 ∨ = H in Figure 5 are not A -cospectral since the A -spectrum of K 2 ∨ = F is { ( − 2 ) [ 2 ] , − 1 , 0 [ 4 ] , 0.4384 , 4.5616 } and the A -spectrum of K 2 ∨ = H is { − 2.6056 , ( − 1 ) [ 2 [ , 0 [ 5 ] , 4.6056 } . The joins F ⊻ K 2 and H ⊻ K 2 in Figure 6 are not A -cospectral since the A -spectrum of F ⊻ K 2 is { − 3.2361 , − 2.5205 , − 1 , − 0.5812 , 0 [ 5 ] , 1.0895 , 1.2361 , 5.0122 } and the A -spectrum of H ⊻ K 2 is { − 3.5337 , − 2.1915 , − 1 , 0 [ 6 ] , 0.3034 , 1.3403 , 5.0815 } , and the joins F ∨ = K 2 and H ∨ = K 2 in Figure 7 are not A -cospectral since the A -spectrum of F ∨ = K 2 is { − 3.1903 , − 2.4142 , − 1.2946 , ( − 1 ) [ 3 ] , 0.4046 , 0.4142 , 1 [ 2 ] , 1.8201 , 5.2602 } and the A -spectrum of H ∨ = K 2 is { − 4.1337 , ( − 1 ) [ 5 ] , 0 , 0.8194 , 1 [ 3 ] , 5.3143 } .

Figure 3

Two nonregular A -cospectral graphs F and H . (a) F and (b) H .

$Figure 4 Two non- A-cospectral \hspace{0.1em}\text{A-cospectral}\hspace{0.1em} graphs K 2 ⊻ F {K}_{2}\hspace{0.33em}⊻\hspace{0.33em}F and K 2 ⊻ H . {K}_{2}\hspace{0.33em}⊻\hspace{0.33em}H. (a) K 2 ⊻ F {K}_{2}\hspace{0.33em}⊻\hspace{0.33em}F and (b) K 2 ⊻ H {K}_{2}\hspace{0.33em}⊻\hspace{0.33em}H .$

Figure 4

Two non- A-cospectral graphs K 2 ⊻ F and K 2 ⊻ H . (a) K 2 ⊻ F and (b) K 2 ⊻ H .

$Figure 5 Two non- A-cospectral \hspace{0.1em}\text{A-cospectral}\hspace{0.1em} graphs K 2 ∨ = F {K}_{2}\mathop{\vee }\limits_{=}F and K 2 ∨ = H {K}_{2}\mathop{\vee }\limits_{=}H . (a) K 2 ∨ = F {K}_{2}\mathop{\vee }\limits_{=}F and (b) K 2 ∨ = H {K}_{2}\mathop{\vee }\limits_{=}H .$

Figure 5

Two non- A-cospectral graphs K 2 ∨ = F and K 2 ∨ = H . (a) K 2 ∨ = F and (b) K 2 ∨ = H .

$Figure 6 Two non- A-cospectral \hspace{0.1em}\text{A-cospectral}\hspace{0.1em} graphs F ⊻ K 2 F\hspace{0.33em}⊻\hspace{0.33em}{K}_{2} and H ⊻ K 2 H\hspace{0.33em}⊻\hspace{0.33em}{K}_{2} . (a) F ⊻ K 2 F\hspace{0.33em}⊻\hspace{0.33em}{K}_{2} and (b) H ⊻ K 2 H\hspace{0.33em}⊻\hspace{0.33em}{K}_{2} .$

Figure 6

Two non- A-cospectral graphs F ⊻ K 2 and H ⊻ K 2 . (a) F ⊻ K 2 and (b) H ⊻ K 2 .

$Figure 7 Two non-A-cospectral graphs F ∨ = K 2 F\mathop{\vee }\limits_{=}{K}_{2} and H ∨ = K 2 H\mathop{\vee }\limits_{=}{K}_{2} . (a) F ∨ = K 2 F\mathop{\vee }\limits_{=}{K}_{2} and (b) H ∨ = K 2 H\mathop{\vee }\limits_{=}{K}_{2} .$

Figure 7

Two non-A-cospectral graphs F ∨ = K 2 and H ∨ = K 2 . (a) F ∨ = K 2 and (b) H ∨ = K 2 .

Remark 3.8

Numerical computations suggest that in the Corollaries 3.4 and 3.5, { A , Q , L } can be replaced by { A , Q , L , ℒ } .

Conjecture 3.9

Let H 1 and H 2 be regular nonisomorphic cospectral graphs. Then, for every G , G ⊻ H 1 and G ⊻ H 2 are { A , Q , L , ℒ } NICS and G ∨ = H 1 and G ∨ = H 2 are { A , Q , L , ℒ } NICS.
Let G 1 and G 2 be regular nonisomorphic cospectral graphs. Then, for every H , G 1 ⊻ H , and G 2 ⊻ H are { A , Q , L , ℒ } NICS and G 1 ∨ = H and G 2 ∨ = H are { A , Q , L , ℒ } NICS.

4 ℒ -Spectra of NS joins

Let G i and H i be r i -regular graphs, i = 1 , 2 . Lu et al. showed that if G 1 and H 1 are cospectral and G 2 and H 2 are cospectral and nonisomorphic, then G 1 ⊻ G 2 and H 1 ⊻ H 2 are { A , L , Q } NICS . In this section, we extend this result by showing that G 1 ⊻ G 2 and H 1 ⊻ H 2 are { A , L , Q , ℒ } NICS . To do it, we determine the spectrum of the normalized Laplacian of the graph G 1 ⊻ G 2 .

Theorem 4.1

Let G 1 be an r 1 -regular graph with n 1 vertices and G 2 be an r 2 -regular graph with n 2 vertices. Then, the normalized Laplacian spectrum of G 1 ⊻ G 2 consists of:

1 + r 2 ( δ i ( G 2 ) − 1 ) n 1 + r 2 for i = 2 , 3 , … , n 2 ;
1 + ( δ i ( G 1 ) − 1 ) ( 9 r 1 2 + 4 r 1 n 2 + r 1 ) 2 ( 2 r 1 + n 2 ) for i = 2 , 3 , … , n 1 ;
1 + ( 1 − δ i ( G 1 ) ) ( 9 r 1 2 + 4 r 1 n 2 − r 1 ) 2 ( 2 r 1 + n 2 ) for i = 2 , 3 , … , n 1 ;
the three roots of the equation
( 2 r 1 r 2 + 2 r 1 n 1 + n 2 r 2 + n 1 n 2 ) x 3 − ( 3 r 1 r 2 + 5 r 1 n 1 + 2 r 2 n 2 + 3 n 1 n 2 ) x 2 + ( 3 r 1 n 1 + n 2 r 2 + 2 n 1 n 2 ) x = 0 .

Proof

Let u 1 , u 2 , … , u n 1 be the vertices of G 1 , u 1 ′ , u 2 ′ , … , u n 1 ′ be the vertices added by the splitting, and v 1 , v 2 , … , v n 2 be the vertices of G 2 . Under this vertex partitioning, the adjacency matrix of G 1 ⊻ G 2 is

A ( G 1 ⊻ G 2 ) = A ( G 1 ) A ( G 1 ) J n 1 × n 2 A ( G 1 ) O n 1 × n 1 O n 1 × n 2 J n 2 × n 1 O n 2 × n 1 A ( G 2 )

The corresponding degrees matrix of G 1 ⊻ G 2 is,

D ( G 1 ⊻ G 2 ) = ( 2 r 1 + n 2 ) I n 1 O n 1 × n 1 O n 1 × n 2 O n 1 × n 1 r 1 I n 1 O n 1 × n 2 O n 2 × n 1 O n 2 × n 1 ( r 2 + n 1 ) I n 2 .

By simple calculation, we obtain

ℒ ( G 1 ⊻ G 2 ) = I n 1 − A ( G 1 ) 2 r 1 + n 2 − A ( G 1 ) r 1 ( 2 r 1 + n 2 ) − J n 1 × n 2 ( 2 r 1 + n 2 ) ( r 2 + n 1 ) − A ( G 1 ) r 1 ( 2 r 1 + n 2 ) I n 1 O n 1 × n 2 − J n 2 × n 1 ( 2 r 1 + n 2 ) ( r 2 + n 1 ) O n 2 × n 1 I n 2 − A ( G 2 ) r 2 + n 1 .

We prove the theorem by constructing an orthogonal basis of eigenvectors of ℒ ( G 1 ⊻ G 2 ) . Since G 2 is r 2 -regular, the vector 1 n 2 is an eigenvector of A ( G 2 ) that corresponds to λ 1 ( G 2 ) = r 2 . For i = 2 , 3 , … , n 2 , let Z i be an eigenvector of A ( G 2 ) that corresponds to λ i ( G 2 ) . Then, 1 n 2 T Z i = 0 and ( 0 1 × n 1 , 0 1 × n 1 , Z i T ) T is an eigenvector of ℒ ( G 1 ⊻ G 2 ) corresponding to the eigenvalue 1 − λ i ( G 2 ) r 2 + n 1 .

By Remark 1.8, 1 + r 2 ( δ i ( G 2 ) − 1 ) n 1 + r 2 are eigenvalues of ℒ ( G 1 ⊻ G 2 ) for i = 2 , … , n 2 .

For i = 2 , … , n 1 , let X i be an eigenvector of A ( G 1 ) corresponding to the eigenvalue λ i ( G 1 ) . We now look for a nonzero real number α such that X i T α X i T 0 1 × n 2 T is an eigenvector of ℒ ( G 1 ⊻ G 2 ) .

ℒ X i α X i 0 n 2 × 1 = X i − λ i ( G 1 ) 2 r 1 + n 2 X i − λ i ( G 1 ) α r 1 ( 2 r 1 + n 2 ) X i − λ i ( G 1 ) r 1 ( 2 r 1 + n 2 ) X i + α X i 0 n 2 × 1 = 1 − λ i ( G 1 ) 2 r 1 + n 2 − λ i ( G 1 ) α r 1 ( 2 r 1 + n 2 ) − λ i ( G 1 ) α r 1 ( 2 r 1 + n 2 ) + 1 0 n 2 × 1 X i α X i 0 n 2 × 1 ,

then, α must be a root of the equation

(4.1) 1 − λ i ( G 1 ) 2 r 1 + n 2 − λ i ( G 1 ) α r 1 ( 2 r 1 + n 2 ) = − λ i ( G 1 ) α r 1 ( 2 r 1 + n 2 ) + 1 ,

2 r 1 + n 2 α 2 + r 1 α − 2 r 1 + n 2 = 0 .

Thus,

α = 2 2 r 1 + n 2 9 r 1 + 4 n 2 + r 1 or α = − 2 2 r 1 + n 2 9 r 1 + 4 n 2 − r 1 .

Substituting the values of α in the right side of equation (4.1), we obtain, by Remark 1.8, that

1 + ( δ i ( G 1 ) − 1 ) ( 9 r 1 2 + 4 r 1 n 2 + r 1 ) n 1 + r 2 , 1 + ( 1 − δ i ( G 1 ) ) ( 9 r 1 2 + 4 r 1 n 2 − r 1 ) 2 ( 2 r 1 + n 2 )

are eigenvalues of ℒ ( G 1 ⊻ G 2 ) for i = 2 , 3 , … , n 1 .

So far, we obtained n 2 − 1 + 2 ( n 1 − 1 ) = 2 n 1 + n 2 − 3 eigenvalues of ℒ ( G 1 ⊻ G 2 ) . Their eigenvectors are orthogonal to ( 1 n 1 T , 0 1 × n 1 , 0 1 × n 2 ) T , ( 0 1 × n 1 , 1 n 1 T , 0 1 × n 2 ) T , and ( 0 1 × n 1 , 0 1 × n 1 , 1 n 2 T ) T .

To find three additional eigenvalues, we look for eigenvectors of ℒ ( G 1 ⊻ G 2 ) of the form Y = ( α 1 n 1 T , β 1 n 1 T , γ 1 n 2 T ) T for ( α , β , γ ) ≠ ( 0 , 0 , 0 ) . Let x be an eigenvalue of ℒ ( G ⊻ G 2 ) corresponding to the eigenvector Y . From ℒ Y = x Y , we obtain

α − r 1 2 r 1 + n 2 α − r 1 r 1 ( 2 r 1 + n 2 ) β − n 2 ( 2 r 1 + n 2 ) ( r 2 + n 1 ) γ = α x − r 1 r 1 ( 2 r 1 + n 2 ) α + β = β x − n 1 ( 2 r 1 + n 2 ) ( r 2 + n 1 ) α + γ − r 2 r 2 + n 1 γ = γ x .

Thus,

α − r 1 2 r 1 + n 2 α + r 1 2 α r 1 ( 2 r 1 + n 2 ) ( x − 1 ) + n 1 n 2 ( r 2 + n 1 ) α ( 2 r 1 + n 2 ) ( r 2 + n 1 ) ( ( x − 1 ) ( r 2 + n 1 ) + r 2 ) = α x .

Note that α ≠ 0 , since if α = 0 , then α = β = γ = 0 , and also x ≠ 1 , since x = 1 implies that α = 0 .

Dividing by α , we obtain the following cubic equation

( 2 r 1 r 2 + 2 r 1 n 1 + n 2 r 2 + n 1 n 2 ) x 3 − ( 3 r 1 r 2 + 5 r 1 n 1 + 2 r 2 n 2 + 3 n 1 n 2 ) x 2 + ( 3 r 1 n 1 + n 2 r 2 + 2 n 1 n 2 ) x = 0 ,

and this completes the proof.□

Now we can answer Question 1.16 by constructing pairs of nonregular { A , L , Q , ℒ } NICS graphs.

Corollary 4.2

Let G i and H i be r i -regular graphs, i = 1 , 2 . If, G 1 and H 1 are cospectral and G 2 and H 2 are cospectral and nonisomorphic then G 1 ⊻ G 2 and H 1 ⊻ H 2 are { A , L , Q , ℒ } NICS.

Proof

G 1 ⊻ G 2 and H 1 ⊻ H 2 are nonisomorphic since G 2 and H 2 are nonisomorphic. By Theorem 4.1 and Theorems 3.1–3.3 in [19], the graphs H 1 ⊻ H 2 are { A , L , Q , ℒ } NICS.□

Example 4.3

Let G 1 = H 1 = C 4 , and if we choose G 2 = G and H 2 = H , where G and H are graphs in Figure 1, then the graphs in Figure 8 are { A , L , Q , ℒ } NICS.

$Figure 8 Nonregular { A , L , Q , ℒ } \{A,L,Q,{\mathcal{ {\mathcal L} }}\} NICS graphs. (a) C 4 ⊻ G 2 {C}_{4}\hspace{0.33em}⊻\hspace{0.33em}{G}_{2} and (b) C 4 ⊻ H 2 {C}_{4}\hspace{0.33em}⊻\hspace{0.33em}{H}_{2} .$

Figure 8

Nonregular { A , L , Q , ℒ } NICS graphs. (a) C 4 ⊻ G 2 and (b) C 4 ⊻ H 2 .

5 Spectra of NNS joins

In this section, we compute the A -spectrum, L -spectrum, Q -spectrum, and ℒ -spectrum of G 1 ∨ = G 2 , where G 1 and G 2 are regular.

We use it to answer Question 1.16 by constructing pairs of nonregular { A , L , Q , ℒ } NICS graphs.

5.1 A-spectra of NNS join

The adjacency matrix of G 1 ∨ = G 2 can be written in a block form

(5.1) A ( G 1 ∨ = G 2 ) = A ( G 1 ) A ( G ¯ 1 ) J n 1 × n 2 A ( G ¯ 1 ) O n 1 × n 1 O n 1 × n 2 J n 2 × n 1 O n 2 × n 1 A ( G 2 ) .

Theorem 5.1

Let G 1 be an r 1 -regular graph with n 1 vertices and G 2 be an r 2 -regular graph with n 2 vertices. Then, the adjacency spectrum of G 1 ∨ = G 2 consists of:

λ j ( G 2 ) for each j = 2 , 3 , … , n 2 ;
two roots of the equation
x 2 − ( λ i ( G 1 ) ) x − ( λ i 2 ( G 1 ) + 2 λ i ( G 1 ) + 1 ) = 0
for each i = 2 , 3 , … , n 1 ;
the three roots of the equation
x 3 − ( r 1 + r 2 ) x 2 + ( r 1 r 2 − ( n 1 − r 1 − 1 ) 2 − n 1 n 2 ) x + r 2 ( n 1 − r 1 − 1 ) 2 = 0

Proof

By Theorem 3.1, the adjacency characteristic polynomial of G 1 ∨ = G 2 is

f A ( G 1 ∨ = G 2 ) ( x ) = x n 1 f A ( G 2 ) ( x ) det x I n 1 − A ( G 1 ) − 1 x A 2 ( G ¯ 1 ) 1 − Γ A ( G 2 ) ( x ) Γ A ( G 1 ) + 1 x A 2 ( G ¯ 1 ) ( x ) = x n 1 ∏ j = 1 n 2 ( x − λ j ( G 2 ) ) det x I n 1 − A ( G 1 ) − 1 x A 2 ( G ¯ 1 ) 1 − Γ A ( G 2 ) ( x ) Γ A ( G 1 ) + 1 x A 2 ( G ¯ 1 ) ( x ) .

Since G 1 and G 2 are regular, we can use Lemma 2.3 to obtain

f A ( G 1 ∨ = G 2 ) ( x ) = x n 1 ∏ j = 1 n 2 ( x − λ j ( G 2 ) ) det x I n 1 − A ( G 1 ) − 1 x A 2 ( G ¯ 1 ) 1 − n 2 x − r 2 n 1 x − r 1 − 1 x ( n 1 − r 1 − 1 ) 2 = x n 1 ∏ j = 1 n 2 ( x − λ j ( G 2 ) ) det x I n 1 − A ( G 1 ) − 1 x ( J − I − A ( G 1 ) ) 2 1 − n 1 n 2 ( x − r 2 ) ( x − r 1 − 1 x ( n 1 − r 1 − 1 ) 2 ) = x n 1 ∏ j = 1 n 2 ( x − λ j ( G 2 ) ) det x I n 1 − A ( G 1 ) − 1 x ( J 2 − 2 J + I − 2 r 1 J + 2 A ( G 1 ) + A 2 ( G 1 ) ) ⋅ 1 − n 1 n 2 ( x − r 2 ) ( x − r 1 − 1 x ( n 1 − r 1 − 1 ) 2 ) = x n 1 ∏ j = 1 n 2 ( x − λ j ( G 2 ) ) det ( B ) − 1 x ( n 1 − 2 − 2 r 1 ) 1 n 1 T adj ( B ) 1 n 1 1 − n 1 n 2 ( x − r 2 ) ( x − r 1 − 1 x ( n 1 − r 1 − 1 ) 2 ) ,

where B = ( x − 1 x ) I n 1 − ( 1 + 2 x ) A ( G 1 ) − 1 x A 2 ( G 1 ) .

Thus, based on Definition 2.2 and Lemma 2.3, we have

f A ( G 1 ∨ = G 2 ) ( x ) = x n 1 ∏ j = 1 n 2 ( x − λ j ( G 2 ) ) ∏ i = 1 n 1 x − 1 x − 1 + 2 x λ i ( G 1 ) − 1 x λ i 2 ( G 1 ) ⋅ 1 − 1 x ( n 1 − 2 − 2 r 1 ) Γ 1 x I n 1 + ( 1 + 2 x ) A ( G 1 ) + 1 x A 2 ( G 1 ) ( x ) 1 − n 1 n 2 ( x − r 2 ) ( x − r 1 − 1 x ( n 1 − r 1 − 1 ) 2 ) = x n 1 ∏ j = 1 n 2 ( x − λ j ( G 2 ) ) ∏ i = 1 n 1 x − 1 x − 1 + 2 x λ i ( G 1 ) − 1 x λ i 2 ( G 1 ) 1 − n 1 − 2 − 2 r 1 x ⋅ n 1 x − 1 x − r 1 − 2 r 1 x − r 1 2 x ⋅ 1 − n 1 n 2 ( x − r 2 ) ( x − r 1 − 1 x ( n 1 − r 1 − 1 ) 2 ) = ∏ j = 1 n 2 ( x − λ j ( G 2 ) ) x n 1 ∏ i = 1 n 1 x − 1 x − 1 + 2 x λ i ( G 1 ) − 1 x λ i 2 ( G 1 ) ⋅ 1 − n 1 ( n 1 − 2 − 2 r 1 ) x 2 − r 1 x − r 1 2 − 2 r 1 − 1 ( x − r 2 ) ( x ( x − r 1 ) − ( n 1 − r 1 − 1 ) 2 ) − n 1 n 2 x ( x − r 2 ) ( x ( x − r 1 ) − ( n 1 − r 1 − 1 ) 2 ) = ∏ j = 2 n 2 ( x − λ j ( G 2 ) ) ∏ i = 2 n 1 ( x 2 − 1 − ( x + 2 ) λ i ( G 1 ) − λ i 2 ( G 1 ) ) ⋅ ( x ( x − r 1 ) − ( n 1 − r 1 − 1 ) 2 ) ( x − r 2 ) ( x ( x − r 1 ) − ( n 1 − r 1 − 1 ) 2 ) − n 1 n 2 x ( x − r 2 ) ( x ( x − r 1 ) − ( n 1 − r 1 − 1 ) 2 ) − n 1 n 2 x = ∏ j = 2 n 2 ( x − λ j ( G 2 ) ) ∏ i = 2 n 1 ( x 2 − 1 − ( x + 2 ) λ i ( G 1 ) − λ i 2 ( G 1 ) ) ⋅ ( ( x − r 2 ) ( x ( x − r 1 ) − ( n 1 − r 1 − 1 ) 2 ) − n 1 n 2 x ) = ∏ j = 2 n 2 ( x − λ j ( G 2 ) ) ∏ i = 2 n 1 ( x 2 − ( λ i ( G 1 ) ) x − ( λ i 2 ( G 1 ) + 2 λ i ( G 1 ) + 1 ) ) ⋅ ( x 3 − ( r 1 + r 2 ) x 2 + ( r 1 r 2 − ( n 1 − r 1 − 1 ) 2 − n 1 n 2 ) x + r 2 ( n 1 − r 1 − 1 ) 2 ) .□

5.2 L-spectra of NNS join

The degrees of the vertices of G 1 ∨ = G 2 are as follows:

d G 1 ∨ = G 2 ( u i ) = n 1 + n 2 − 1 , i = 1 , … , n 1 ,

d G 1 ∨ = G 2 ( u i ′ ) = n 1 − r 1 − 1 , i = 1 , … , n 1 ,

d G 1 ∨ = G 2 ( v j ) = r 2 + n 1 , j = 1 , … , n 2 ,

so the degrees matrix of G 1 ∨ = G 2 that corresponds to equation (5.1) is

(5.2) D ( G 1 ∨ = G 2 ) = ( n 1 + n 2 − 1 ) I n 1 O n 1 × n 1 O n 1 × n 2 O n 1 × n 1 ( n 1 − r 1 − 1 ) I n 1 O O n 2 × n 1 O n 2 × n 1 ( r 2 + n 1 ) I n 2

Theorem 5.2

Let G 1 be an r 1 -regular graph with n 1 vertices and G 2 be an r 2 -regular graph with n 2 vertices. Then, the Laplacian spectrum of G 1 ∨ = G 2 consists of the following:

n 1 + μ j ( G 2 ) for each j = 2 , 3 , … , n 2 ;
two roots of the equation
x 2 + ( 2 r 1 − 2 n 1 − n 2 − μ i ( G 1 ) + 2 ) x + n 1 2 − 2 r 1 n 1 − 2 n 1 + n 1 n 2 − r 1 n 2 − n 2 + μ i ( G 1 ) ( n 1 + r 1 + 1 − μ i ( G 1 ) ) = 0
for each i = 2 , 3 , … , n 1 ;
the three roots of the equation
x 3 + ( 2 r 1 − 3 n 1 − n 2 + 2 ) x 2 + ( n 1 n 2 − n 2 r 1 − n 2 + 2 n 1 2 − 2 r 1 n 1 − 2 n 1 ) x = 0 .

Proof

By substituting D ( G 1 ) = r 1 I n 1 in Theorem 3.2, the Laplacian characteristic polynomial of G 1 ∨ = G 2 is

f L ( G 1 ∨ = G 2 ) ( x ) = det ( ( x − n 1 ) I n 2 − L ( G 2 ) ) det ( x − n 1 + 1 + r 1 ) I n 1 ⋅ det ( x − n 1 − n 2 + r 1 + 1 ) I n 1 − L ( G 1 ) − 1 x − n 1 + r 1 + 1 A 2 ( G ¯ 1 ) ⋅ 1 − Γ L ( G 2 ) ( x − n 1 ) Γ L ( G 1 ) + 1 x − n 1 + 1 + r 1 A 2 ( G ¯ 1 ) ( x − n 1 − n 2 + 1 + r 1 ) .

Using Lemma 2.3, we obtain

f L ( G 1 ∨ = G 2 ) ( x ) = det ( ( x − n 1 ) I n 2 − L ( G 2 ) ) ( x − n 1 + r 1 + 1 ) n 1 ⋅ det ( x − n 1 − n 2 + r 1 + 1 ) I n 1 − L ( G 1 ) − 1 x − n 1 + r 1 + 1 A 2 ( G ¯ 1 ) ⋅ 1 − n 2 n 1 ( x − n 1 ) x − n 1 − n 2 + 1 + r 1 − ( n 1 − r 1 − 1 ) 2 x − n 1 + r 1 + 1 = ∏ j = 1 n 2 ( x − n 1 − μ j ( G 2 ) ) ( x − n 1 + r 1 + 1 ) n 1 ( x − n 1 − n 2 + 1 + r 1 ) − ( n 1 − r 1 − 1 ) 2 x − n 1 + r 1 + 1 ⋅ ∏ i = 2 n 1 x − n 1 − n 2 + 1 + r 1 − μ i ( G 1 ) − ( μ i ( G 1 ) − r 1 − 1 ) 2 x − n 1 + r 1 + 1 ⋅ 1 − n 2 n 1 ( x − n 1 ) x − n 1 − n 2 + 1 + r 1 − ( n 1 − r 1 − 1 ) 2 x − n 1 + r 1 + 1 = ∏ j = 2 n 2 ( x − n 1 − μ j ( G 2 ) ) ∏ i = 2 n 1 ( x 2 + ( 2 r 1 − 2 n 1 − n 2 − μ i ( G 1 ) + 2 ) x + n 1 2 − 2 r 1 n 1 − 2 n 1 + n 1 n 2 − r 1 n 2 − n 2 + μ i ( G 1 ) ( n 1 + r 1 + 1 − μ i ( G 1 ) ) ) ⋅ [ x 3 + ( 2 r 1 − 3 n 1 − n 2 + 2 ) x 2 + ( n 1 n 2 − n 2 r 1 − n 2 + 2 n 1 2 − 2 r 1 n 1 − 2 n 1 ) x ] .

This completes the proof.□

5.3 Q-spectra of NNS join

Theorem 5.3

Let G 1 be an r 1 -regular graph with n 1 vertices and G 2 be an r 2 -regular graph with n 2 vertices. Then, the signless Laplacian spectrum of G 1 ∨ = G 2 consists of the following:

n 1 + ν j ( G 2 ) for each j = 2 , 3 , … , n 2 ;
two roots of the equation x 2 + ( 2 r 1 − 2 n 1 − n 2 − ν i ( G 1 ) + 2 ) x + n 1 2 − 2 r 1 n 1 − 2 n 1 + n 1 n 2 − r 1 n 2 − n 2 + 4 r 1 + ν i ( G 1 ) ( n 1 + r 1 − 3 − ν i ( G 1 ) ) = 0 for each i = 2 , 3 , … , n 1 ;
the three roots of the equation
x 3 + ( 2 − 3 n 1 − n 2 − 2 r 2 ) x 2 + ( n 1 n 2 − n 2 r 1 − n 2 + 2 n 1 2 + 2 r 1 n 1 − 2 n 1 − 2 r 1 − 2 r 1 2 + 4 r 2 n 1 − 4 r 2 + 2 r 2 n 2 ) x + 2 n 1 r 1 2 + 2 r 1 n 1 − 2 r 1 n 1 2 − 2 r 2 n 1 n 2 + 2 r 1 r 2 n 2 + 2 r 2 n 2 + 4 r 2 r 1 2 + 4 r 1 r 2 − 4 r 1 r 2 n 1 = 0

Proof

The proof is similar to the proof of Theorem 5.2.□

5.4 ℒ -spectra of NNS join

Let G 1 be an r 1 -regular graph on order n 1 . Let S be a subset of { 2 , 3 , … , n 1 } such that δ i ( G 1 ) = 1 + 1 r 1 for i ∈ S and denote the cardinality of S by n ( S ) . Let G 2 be an r 2 -regular graph on order n 2 . In the following theorem, we determine the normalized Laplacian spectrum of G 1 ∨ = G 2 in terms of the normalized Laplacian eigenvalues of G 1 and G 2 . The proof is slightly more complicated than the proof of Theorem 4.1, and we consider three cases.

Theorem 5.4

If S = Φ , then the normalized Laplacian spectrum of G 1 ∨ = G 2 consists of the following:
1. 1 + r 2 ( δ i ( G 2 ) − 1 ) n 1 + r 2 for each i = 2 , 3 , … , n 2 ;
2. 1 + 2 ( 1 + r 1 − r 1 δ i ( G 1 ) ) 2 r 1 ( 1 − δ i ( G 1 ) ) ( n 1 − r 1 − 1 ) ∓ ( n 1 − r 1 − 1 ) [ r 1 2 ( 1 − δ i ( G 1 ) ) 2 ( n 1 − r 1 − 1 ) + 4 ( 1 + r 1 − r 1 δ i ( G 1 ) ) 2 ( n 1 + n 2 − 1 ) ] for each i = 2 , 3 , … , n 1 ;
3. the three roots of the equation
  ( n 1 2 + n 1 n 2 − n 1 + r 2 n 1 + r 2 n 2 − r 2 ) x 3 − ( 3 n 1 2 + 3 n 1 n 2 − 3 n 1 − r 1 n 1 + 2 r 2 n 1 + 2 r 2 n 2 − 2 r 2 − r 1 r 2 ) x 2 + ( 2 n 1 2 + 2 n 1 n 2 − 2 n 1 − r 1 n 1 + r 2 n 2 ) x = 0 .
If S = { 2 , 3 , … , n 1 } , then the normalized Laplacian spectrum of G 1 ∨ = G 2 consists of the following:
1. 1 + r 2 ( δ i ( G 2 ) − 1 ) n 1 + r 2 for each i = 2 , 3 , … , n 2 ;
2. ( 1 + 1 n 1 + n 2 − 1 ) [ n 1 − 1 ] , 0 [ n 1 + 1 ] , and n 1 2 + 2 n 1 n 2 + r 2 n 2 − n 1 ( r 2 + n 1 ) ( n 1 + n 2 − 1 ) .
If S ≠ Φ and S ≠ { 2 , 3 , … , n 1 } , then the normalized Laplacian spectrum of G 1 ∨ = G 2 consists of the following:
1. 1 + r 2 ( δ i ( G 2 ) − 1 ) n 1 + r 2 for each i = 2 , 3 , … , n 2 ;
2. 1 + 2 ( 1 + r 1 − r 1 δ i ( G 1 ) ) 2 r 1 ( 1 − δ i ( G 1 ) ) ( n 1 − r 1 − 1 ) ∓ ( n 1 − r 1 − 1 ) [ r 1 2 ( 1 − δ i ( G 1 ) ) 2 ( n 1 − r 1 − 1 ) + 4 ( 1 + r 1 − r 1 δ i ( G 1 ) ) 2 ( n 1 + n 2 − 1 ) ] for each i ∈ { 2 , 3 , … , n 1 } ⧹ S ;
3. 1 [ n ( S ) ] and ( 1 + 1 n 1 + n 2 − 1 ) [ n ( S ) ] .
4. the three roots of the equation
  ( n 1 2 + n 1 n 2 − n 1 + r 2 n 1 + r 2 n 2 − r 2 ) x 3 − ( 3 n 1 2 + 3 n 1 n 2 − 3 n 1 − r 1 n 1 + 2 r 2 n 1 + 2 r 2 n 2 − 2 r 2 − r 1 r 2 ) x 2 + ( 2 n 1 2 + 2 n 1 n 2 − 2 n 1 − r 1 n 1 + r 2 n 2 ) x = 0 .

Proof

(a) If S = Φ , then δ i ( G 1 ) ≠ 1 + 1 r 1 for each i = 2 , 3 , … , n 1 , so λ i ( G 1 ) ≠ − 1 for each i = 2 , 3 , … , n 1 . The normalized Laplacian matrix of G 1 ∨ = G 2 is:

ℒ ( G 1 ∨ = G 2 ) = I n 1 − A ( G 1 ) n 1 + n 2 − 1 − A ( G ¯ 1 ) ( n 1 + n 2 − 1 ) ( n 1 − r 1 − 1 ) − J n 1 × n 2 ( n 1 + n 2 − 1 ) ( r 2 + n 1 ) − A ( G ¯ 1 ) ( n 1 + n 2 − 1 ) ( n 1 − r 1 − 1 ) I n 1 O n 1 × n 2 − J n 2 × n 1 ( n 1 + n 2 − 1 ) ( r 2 + n 1 ) O n 2 × n 1 I n 2 − A ( G 2 ) r 2 + n 1 .

Since G 2 is r 2 -regular, the vector 1 n 2 is an eigenvector of A ( G 2 ) that corresponds to λ 1 ( G 2 ) = r 2 . For i = 2 , 3 , … , n 2 let Z i be an eigenvector of A ( G 2 ) that corresponds to λ i ( G 2 ) . Then, 1 n 2 T Z i = 0 and ( 0 1 × n 1 , 0 1 × n 1 , Z i T ) T is an eigenvector of ℒ ( G 1 ∨ = G 2 ) corresponding to the eigenvalue 1 − λ i ( G 2 ) r 2 + n 1 . By Remark 1.8, 1 + r 2 ( δ i ( G 2 ) − 1 ) n 1 + r 2 are an eigenvalues of L ( G 1 ∨ = G 2 ) for i = 2 , … , n 2 .

For i = 2 , … , n 1 , let X i be an eigenvector of A ( G 1 ) corresponding to the eigenvalue λ i ( G 1 ) . We now look for a nonzero real number α such that X i T α X i T 0 1 × n 2 T is an eigenvector of ℒ ( G 1 ∨ = G 2 ) . Note that α ≠ 0 , since if α = 0 , then λ i ( G 1 ) = − 1 .

ℒ X i α X i 0 n 1 × 1 = X i − λ i ( G 1 ) n 1 + n 2 − 1 X i + 1 + λ i ( G 1 ) ( n 1 + n 2 − 1 ) ( n 1 − r 1 − 1 ) α X i 1 + λ i ( G 1 ) ( n 1 + n 2 − 1 ) ( n 1 − r 1 − 1 ) X i + α X i 0 n 1 × 1 = 1 − λ i ( G 1 ) n 1 + n 2 − 1 + 1 + λ i ( G 1 ) ( n 1 + n 2 − 1 ) ( n 1 − r 1 − 1 ) α 1 + λ i ( G 1 ) α ( n 1 + n 2 − 1 ) ( n 1 − r 1 − 1 ) + 1 0 n 1 × 1 X i α X i 0 n 1 × 1 .

Thus,

(5.3) 1 − λ i ( G 1 ) n 1 + n 2 − 1 + 1 + λ i ( G 1 ) ( n 1 + n 2 − 1 ) ( n 1 − r 1 − 1 ) α = 1 + λ i ( G 1 ) α ( n 1 + n 2 − 1 ) ( n 1 − r 1 − 1 ) + 1

− λ i ( G 1 ) n 1 + n 2 − 1 + 1 + λ i ( G 1 ) ( n 1 + n 2 − 1 ) ( n 1 − r 1 − 1 ) α = 1 + λ i ( G 1 ) α ( n 1 + n 2 − 1 ) ( n 1 − r 1 − 1 )

α 2 ( 1 + λ i ( G 1 ) ) − ( 1 + λ i ( G 1 ) ) α ( n 1 + n 2 − 1 ) ( n 1 − r 1 − 1 ) = λ i ( G 1 ) n 1 + n 2 − 1

( 1 + λ i ( G 1 ) ) n 1 + n 2 − 1 α 2 − λ i ( G 1 ) n 1 − r 1 − 1 α − ( 1 + λ i ( G 1 ) ) n 1 + n 2 − 1 = 0 ,

α 1 , 2 = λ i ( G 1 ) n 1 − r 1 − 1 2 ( 1 + λ i ( G 1 ) ) n 1 + n 2 − 1 ∓ λ i ( G 1 ) 2 ( n 1 − r 1 − 1 ) 4 ( 1 + λ i ( G 1 ) ) 2 ( n 1 + n 2 − 1 ) + 1 .

Substituting the values of α in the right side of equation (5.3), we obtain by Remark 1.8 that

1 + 2 ( 1 + r 1 − r 1 δ i ( G 1 ) ) 2 r 1 ( 1 − δ i ( G 1 ) ) ( n 1 − r 1 − 1 ) ∓ ( n 1 − r 1 − 1 ) [ r 1 2 ( 1 − δ i ( G 1 ) ) 2 ( n 1 − r 1 − 1 ) + 4 ( 1 + r 1 − r 1 δ i ( G 1 ) ) 2 ( n 1 + n 2 − 1 ) ]

are eigenvalues of ℒ ( G 1 ∨ = G 2 ) for each i = 2 , 3 , … , n 1 .

So far, we have obtained n 2 − 1 + 2 ( n 1 − 1 ) = 2 n 1 + n 2 − 3 eigenvalues of ℒ ( G 1 ∨ = G 2 ) . The corresponding eigenvectors are orthogonal to ( 1 n i T , 0 1 × n 1 , 0 1 × n 2 ) T , ( 0 1 × n 1 , 1 n i T , 0 1 × n 2 ) T and ( 0 1 × n 1 , 0 1 × n 1 , 1 n 2 T ) T . To find three additional eigenvalues, we look for eigenvectors of ℒ ( G 1 ∨ = G 2 ) of the form Y = ( α 1 n 1 T , β 1 n 1 T , γ 1 n 2 T ) T for ( α , β , γ ) ≠ ( 0 , 0 , 0 ) . Let x be an eigenvalue of ℒ ( G ∨ = G 2 ) corresponding to the eigenvector Y . From ℒ Y = x Y , we obtain

(5.4) α − α r 1 n 1 + n 2 − 1 + ( 1 + r 1 − n 1 ) β n 1 + n 2 − 1 n 1 − r 1 − 1 − n 2 γ ( n 1 + n 2 − 1 ) ( r 2 + n 1 ) = α x

(5.5) ( 1 + r 1 − n 1 ) α n 1 − r 1 − 1 n 1 + n 2 − 1 + β = β x

(5.6) − n 1 α r 2 + n 1 n 1 + n 2 − 1 + γ − r 2 γ r 2 + n 1 γ x

Thus,

α − α r 1 n 1 + n 2 − 1 + α ( n 1 − 1 − r 1 ) ( n 1 + n 2 − 1 ) ( x − 1 ) + α ( n 1 n 2 ) ( n 1 + n 2 − 1 ) ( x r 2 + n 1 ( x − 1 ) ) = α x .

Note that α ≠ 0 , since if α = 0 , then α = β = γ = 0 , and also x ≠ 1 , since x = 1 implies that α = 0 .

Dividing by α , we obtain

1 − r 1 n 1 + n 2 − 1 + n 1 − 1 − r 1 ( n 1 + n 2 − 1 ) ( x − 1 ) + n 1 n 2 ( n 1 + n 2 − 1 ) ( x r 2 + n 1 ( x − 1 ) ) = x .

Then,

( n 1 2 + n 1 n 2 − n 1 ) ( x − 1 ) 3 + ( r 2 n 1 x + r 2 n 2 x − r 2 x + r 1 n 1 ) ( x − 1 ) 2 + ( r 1 r 2 x − n 1 2 + n 1 + r 1 n 1 − n 1 n 2 ) ( x − 1 ) − r 2 ( n 1 − 1 − r 1 ) x = 0 ,

then, by simple calculation, we see that x is a root of the cubic equation

( n 1 2 + n 1 n 2 − n 1 + r 2 n 1 + r 2 n 2 − r 2 ) x 3 − ( 3 n 1 2 + 3 n 1 n 2 − 3 n 1 − r 1 n 1 + 2 r 2 n 1 + 2 r 2 n 2 − 2 r 2 − r 1 r 2 ) x 2 + ( 2 n 1 2 + 2 n 1 n 2 − 2 n 1 − r 1 n 1 + r 2 n 2 ) x = 0 ,

and this completes the proof of (a).

(b) The proof of (i) is similar to the proof of (i) in (a). Now we prove (ii). If S = { 2 , 3 , … , n 1 } , then δ j ( G 1 ) = 1 + 1 r 1 for each i = 2 , 3 , … , n 1 , so λ i ( G 1 ) = − 1 for each i = 2 , 3 , … , n 1 , i.e., G 1 = K n 1 and r 1 = n 1 − 1 . So the normalized Laplacian matrix of G 1 ∨ = G 1 is as follows:

ℒ ( G 1 ∨ = G 2 ) = I n 1 − A ( G 1 ) n 1 + n 2 − 1 O n 1 × n 1 − J n 1 × n 2 ( n 1 + n 2 − 1 ) ( r 2 + n 1 ) O n 1 × n 1 O n 1 × n 1 O n 1 × n 2 − J n 2 × n 1 ( n 1 + n 2 − 1 ) ( r 2 + n 1 ) O n 2 × n 1 I n 2 − A ( G 2 ) r 2 + n 1 .

For i = 2 , … , n 1 , let X i be an eigenvector of A ( G 1 ) corresponding to the eigenvalue λ i ( G 1 ) = − 1 . So X i T 0 1 × n 1 0 1 × n 2 T is an eigenvector of ℒ ( G 1 ∨ = G 2 ) corresponding to the eigenvalue 1 + 1 n 1 + n 2 − 1 and 0 1 × n 1 X i T 0 1 × n 2 T is an eigenvector of ℒ ( G 1 ∨ = G 2 ) corresponding to the eigenvalue 0 because,

ℒ X i 0 n 1 × 1 0 n 2 × 1 = X i − λ i ( G 1 ) X i n 1 + n 2 − 1 0 n 1 × 1 0 n 2 × 1 = 1 + 1 n 1 + n 2 − 1 X i 0 n 1 × 1 0 n 2 × 1 and ℒ 0 n 1 × 1 X i 0 n 2 × 1 = 0 n 1 × 1 0 n 1 × 1 0 n 2 × 1 .

Therefore, ( 1 + 1 n 1 + n 2 − 1 ) [ n 1 − 1 ] and 0 [ n 1 − 1 ] are eigenvalues of ℒ ( G 1 ∨ = G 2 ) .

So far, we have obtained n 2 − 1 + 2 ( n 1 − 1 ) = 2 n 1 + n 2 − 3 eigenvalues of ℒ ( G 1 ∨ = G 2 ) . Their eigenvectors are orthogonal to ( 1 n 1 T , 0 1 × n 1 , 0 1 × n 2 ) T , ( 0 1 × n 1 , 1 n 1 T , 0 1 × n 2 ) T , and ( 0 1 × n 1 , 0 1 × n 1 , 1 n 2 T ) T . To find three additional eigenvalues, we look for eigenvectors of ℒ ( G ∨ = G 2 ) of the form Y = ( α 1 n 1 T , β 1 n 1 T , γ 1 n 2 T ) T for ( α , β , γ ) ≠ ( 0 , 0 , 0 ) . Let x be an eigenvalue of ℒ ( G ∨ = G 2 ) corresponding to the eigenvector Y . Then, from ℒ Y = x Y , we obtain

(5.7) α − r 1 α n 1 + n 2 − 1 − γ n 2 ( n 1 + n 2 − 1 ) ( r 2 + n 1 ) = α x

(5.8) β x = 0

(5.9) − α n 1 ( n 1 + n 2 − 1 ) ( r 2 + n 1 ) + γ − r 2 γ r 2 + n 1 = γ x

If β ≠ 0 , then ( 0 , β , 0 ) is a one of the solutions of the above three equations, so ( 0 1 × n 1 , β 1 n 1 T , 0 1 × n 2 ) T is an eigenvector corresponding to the eigenvalue 0. On the other hand, if β = 0 , we obtain

(5.10) α 1 − x − r 1 n 1 + n 2 − 1 = γ n 2 ( n 1 + n 2 − 1 ) ( r 2 + n 1 )

(5.11) γ 1 − x − r 2 r 2 + n 1 = α n 1 ( n 1 + n 2 − 1 ) ( r 2 + n 1 ) .

By solving the above two equations, we obtain the following equation:

( r 2 + n 1 ) ( n 1 + n 2 − 1 ) x 2 + ( n 1 − r 2 n 2 − 2 n 1 n 2 − n 1 2 ) x = 0 ,

whose roots are 0 and n 1 2 + 2 n 1 n 2 + r 2 n 2 − n 1 ( r 2 + n 1 ) ( n 1 + n 2 − 1 ) .

This completes the proof of (b).

(c) The proofs of (i), (ii), and (iv) are similar to the proofs of (i), (ii) and (iii) of (a), respectively. Now we prove (iii), Let S ≠ Φ and S ≠ { 2 , 3 , … , n 1 } . If i ∈ S , then 1 + 1 n 1 + n 2 − 1 and 1 are eigenvalues of ℒ ( G ∨ = G 2 ) because if X i is an eigenvector corresponding to the eigenvalue δ i ( G 1 ) , then

ℒ X i 0 n 1 × 1 0 n 2 × 1 = X i + X i n 1 + n 2 − 1 0 n 1 × 1 0 n 2 × 1 = 1 + 1 n 1 + n 2 − 1 X i 0 n 1 × 1 0 n 2 × 1 and ℒ 0 n 1 × 1 X i 0 n 2 × 1 = 0 n 1 × 1 X i 0 n 2 × 1 .

So, 1 [ n ( S ) ] and ( 1 + 1 n 1 + n 2 − 1 ) [ n ( S ) ] are eigenvalues of ℒ ( G ∨ = G 2 ) , and this completes the proof of (c).□

Now we can give another answer to Question 1.16 by constructing several pairs of nonregular { A , L , Q , ℒ } NICS graphs.

Corollary 5.5

Let G 1 and H 1 be cospectral regular graphs and G 2 and H 2 be nonisomorphic, regular, and cospectral graphs. Then, G 1 ∨ = G 2 and H 1 ∨ = H 2 are nonregular { A , L , Q , ℒ } NICS.

Proof

G 1 ∨ = G 2 and H 1 ∨ = H 2 are nonisomorphic since G 2 and H 2 are nonisomorphic. By Theorems 5.1–5.4, we obtain that G 1 ∨ = G 2 and H 1 ∨ = H 2 are nonregular { A , L , Q , ℒ } NICS.□

Example 5.6

Let G 1 = H 1 = C 4 , and if we choose G 2 = G and H 2 = H , where G and H are graphs in Figure 1, then the graphs in Figure 9 are { A , L , Q , ℒ } NICS.

$Figure 9 Non regular { A , L , Q , ℒ } \{A,L,Q,{\mathcal{ {\mathcal L} }}\} NICS graphs. (a) C 4 ∨ = G 2 {C}_{4}\mathop{\vee }\limits_{=}{G}_{2} and (b) C 4 ∨ = H 2 {C}_{4}\mathop{\vee }\limits_{=}{H}_{2} .$

Figure 9

Non regular { A , L , Q , ℒ } NICS graphs. (a) C 4 ∨ = G 2 and (b) C 4 ∨ = H 2 .

Dedicated to Prof. Frank J. Hall, in celebration of his many contributions to matrix theory.

Acknowledgements

The authors are thankful to the editors and the referee for careful reading of the article and the encouraging comments made in the report.

Conflict of interest: The authors state no conflict of interest.
Data availability statement: Data available within the article or from the corresponding author on reasonable request.

References

[1] S. Barik, R. B Bapat, and S. Pati, On the Laplacian spectra of product graphs, Appl. Anal. Discrete Math. 9 (2015), 39–58. Search in Google Scholar

[2] S. Barik, S. Pati, and B. K. Sarma, The spectrum of the corona of two graphs, SIAM J. Discrete Math. 21 (2007), no. 1, 47–56. Search in Google Scholar

[3] A. E. Brouwer and W. H. Haemers, Spectra of Graphs, Springer Science & Business Media, New York, 2011. Search in Google Scholar

[4] S. Butler, A note about cospectral graphs for the adjacency and normalized Laplacian matrices, Linear Multilinear Algebra 58 (2010), no. 3, 387–390. Search in Google Scholar

[5] D. M. Cardoso, M. A. A. de Freitas, E. A. Martins, and M. Robbiano, Spectra of graphs obtained by a generalization of the join graph operation, Discrete Math. 313 (2013), no. 5, 733–741. Search in Google Scholar

[6] F. R. K. Chung and F. Chung Graham, Spectral Graph Theory, vol 92, American Mathematical Society, Rhode Island, 1997. Search in Google Scholar

[7] S.-Y. Cui and G.-X. Tian, The spectrum and the signless Laplacian spectrum of coronae, Linear Algebra Appl. 437 (2012), no. 7, 1692–1703. Search in Google Scholar

[8] D. Cvetkovič, P. Rowlinson, and S. Simić, An Introduction to the Theory of Graph Spectra London Mathematical Society Student Texts, Cambridge University, London, 2010. Search in Google Scholar

[9] D. Cvetkovic, Spectra of graphs formed by some unary operations, Publ. Inst. Math. (Beograd) 19 (1975), no. 33, 37–41. Search in Google Scholar

[10] D. M. Cvetković, Graphs and their spectra, Publikacije Elektrotehničkog fakulteta. Serija Matematika i fizika (1971), no. 354/356, 1–50. Search in Google Scholar

[11] A. Das and P. Panigrahi, New classes of simultaneous cospectral graphs for adjacency, Laplacian and normalized Laplacian matrices, Kragujevac J. Math. 43 (2019), no. 2, 303–323. Search in Google Scholar

[12] C. D. Godsil and B. D. McKay, A new graph product and its spectrum, Bulletin Aust. Math. Soc. 18 (1978), no. 1, 21–28. Search in Google Scholar

[13] I. Gopalapillai, The spectrum of neighborhood corona of graphs, Kragujevac J. Math. 35 (2011), no. 3, 493–500. Search in Google Scholar

[14] F. Harary, Graph Theory, Addison Wesley Publishing Company, Reading, Massachusetts, 1969. Search in Google Scholar

[15] R. A. Horn and F. Zhang, Basic properties of the Schur complement, The Schur Complement and Its Applications 4 (2005), 17–46. Search in Google Scholar

[16] Y. Hou and W.-C. Shiu, The spectrum of the edge corona of two graphs, Electr. J. Linear Algebra 20 (2010), 586–594. Search in Google Scholar

[17] G. Indulal, Spectrum of two new joins of graphs and infinite families of integral graphs, Kragujevac J. Math. 36 (2012), no. 38, 133–139. Search in Google Scholar

[18] X. Liu and Z. Zhang, Spectra of subdivision-vertex join and subdivision-edge join of two graphs, Bullet. Malaysian Math. Sci. Soc. 42 (2019), no. 1, 15–31. Search in Google Scholar

[19] Z. Lu, X. Ma, and M. Zhang, Spectra of graph operations based on splitting graph, J. Appl. Anal. Comput. 13 (2023), no. 1, 133–155. Search in Google Scholar

[20] C. McLeman and E. McNicholas, Spectra of Coronae, Linear Algebra Appl. 435 (2011), no. 5, 998–1007. Search in Google Scholar

[21] B. Mohar, Laplace eigenvalues of graphs – a survey, Discrete Math. 109 (1992), no. 1–3, 171–183. Search in Google Scholar

[22] B. Nica, A brief introduction to spectral graph theory, (2016), arXiv: http://arXiv.org/abs/arXiv:1609.08072. Search in Google Scholar

[23] R. Pavithra and R. Rajkumar, Spectra of m-edge rooted product of graphs, Indian J. Pure Appl. Math. 52 (2021), no. 4, 1235–1255. Search in Google Scholar

[24] R. Pavithra and R. Rajkumar, Spectra of bowtie product of graphs, Discrete Math. Algorithms Appl. 14 (2022), no. 02, 2150114. Search in Google Scholar

[25] R. Rajkumar and M. Gayathri, Spectra of (h1, h2)-merged subdivision graph of a graph, Indag Math. 30 (2019), no. 6, 1061–1076. Search in Google Scholar

[26] R. Rajkumar and R. Pavithra, Spectra of m-rooted product of graphs, Linear Multilinear Algebra 70 (2022), no. 1, 1–26. Search in Google Scholar

[27] D. A. Spielman, Spectral graph theory and its applications, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS’07), IEEE, 2007, pp. 29–38. Search in Google Scholar

[28] E. R. Van Dam and W. H. Haemers, Which graphs are determined by their spectrum?, Linear Algebra Appl. 373 (2003), 241–272. Search in Google Scholar

[29] E. R. Van Dam and W. H. Haemers, Developments on spectral characterizations of graphs, Discrete Math. 309 (2009), no. 3, 576–586. Search in Google Scholar

[30] R. P. Varghese and D. Susha, On the normalized Laplacian spectrum of some graphs, Kragujevac J. Math. 44 (2020), no. 3, 431–442. Search in Google Scholar

Received: 2023-08-07

Revised: 2023-11-07

Accepted: 2023-11-10

Published Online: 2024-02-22

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/spma-2023-0109

Keywords for this article

spectra of graphs; nonisomorphic cospectral graphs; NS and NNS joins; coronal and Schur complement

Creative Commons

BY 4.0