Least eigenvalue of the connected graphs whose complements are cacti

Haiying Wang; Muhammad Javaid; Sana Akram; Muhammad Jamal; Shaohui Wang

doi:10.1515/math-2019-0097

Article Open Access

Least eigenvalue of the connected graphs whose complements are cacti

Haiying Wang , Muhammad Javaid , Sana Akram , Muhammad Jamal and Shaohui Wang

Published/Copyright: November 28, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

Suppose that Γ is a graph of order n and A(Γ) = [a_i,j] is its adjacency matrix such that a_i,j is equal to 1 if v_i is adjacent to v_j and a_i,j is zero otherwise, where 1 ≤ i, j ≤ n. In a family of graphs, a graph is called minimizing if the least eigenvalue of its adjacency matrix is minimum in the set of the least eigenvalues of all the graphs. Petrović et al. [On the least eigenvalue of cacti, Linear Algebra Appl., 2011, 435, 2357-2364] characterized a minimizing graph in the family of all cacti such that the complement of this minimizing graph is disconnected. In this paper, we characterize the minimizing graphs G ∈ Ωnc , i.e.

λmin(G)≤λmin(Cc)

for each C^c ∈ Ωnc , where Ωnc is a collection of connected graphs such that the complement of each graph of order n is a cactus with the condition that either its each block is only an edge or it has at least one block which is an edge and at least one block which is a cycle.

Keywords: adjacency matrix; least eigenvalue; connected graphs; cacti

MSC 2010: 15A18; 05C50; 05C40; 05D05

1 Introduction

Let Γ = (V(Γ), E(Γ)) be a graph such that V(Γ) = {v_i : 1 ≤ i ≤ n} and E(Γ) are set of vertices and edges respectively. Assume that all the considered graphs are simple, finite and undirected. For each i, the degree d(i) is the number of incident edges on v_i. The adjacency matrix of Γ is A(Γ) = [a_i,j] with a_i,j equal to 1 if v_i is linked to v_j and a_i,j is zero for the rest case, where 1 ≤ i, j ≤ n. The solutions of det(A(Γ) – λI) = 0 are eigenvalues of Γ. It is interesting to note that A(Γ) is always symmetric and real, all the eigenvalues can be arrange as λ₁(Γ) ≤ λ₂(Γ) ≤ … ≤ λ_n(Γ). The eigenvectors corresponding to the least eigenvalue λ₁(Γ) and the greatest eigenvalue (spectral radius) λ_n(Γ) are called first eigenvector (FEv) and Perron-vector respectively.

The spectrum of the adjacency matrix for an undirected graph is first time studied by Collatz and Sinogowitz (1957), see [1]. Later on, many researchers discussed the largest eigenvalue (spectral radius) in the area of spectra of graphs, see [2, 3]. It is observed that the least eigenvalue did not receive the attention of researchers as compare of the largest eigenvalue. From the few of results of the least eigenvalues on the graphs, the bounds related results can be found in [4, 5]. For further study, we refer [6, 7, 8, 9, 10, 11, 12, 13]. A graph Γ is said to be minimizing in a certain collection of graphs, if the least eigenvalue of A(Γ) is minimum in the set of all the least eigenvalues of the other graphs in the same collection.

Let 𝓖(p, q) be a collection of connected graphs in which each graph is of p order and q size such that 0<q<p(p−1)2. The minimizing graph in 𝓖(p, q) characterized by Bell et al. [14] is stated in the below result:

Theorem 1.1

Minimizing graph in 𝓖(p, q) is either a join of two nested split graphs, or a bipartite graph.

It is important to note that the complements of the graphs characterized by Bell contain the cliques such that order of each clique is greater or equal to p2 or these are disconnected. After it, the question is raised to investigate the minimizing graphs in the collection of connected graphs such that the complement of each graph contains the cliques of small sizes. Motivated by it, the minimizing graph in the collection of connected graphs such that the complement of each graph is trees, unicyclic or bicyclic are characterized by Fan, Zhang, Wang, Li and Javaid, see [15, 16, 17, 18]. For further study, we refer to [19, 20, 21, 22]. In this paper, the minimizing graph is characterized in the collection of connected graphs such that the complement of each graph is cactus with the condition that each block of a cactus of order n is only an edge or a cactus of order n has at least one block which is an edge and at least one block which is a cycle. In the rest of paper; Section 2 includes some basic definitions and terminologies, Section 3 contains the proofs of some important lemmas and Section 4 has the main results in which minimizing graph is characterized in the family of connected graphs with the condition that the complement of each graph is cactus.

2 Preliminaries

A connected graph is called cactus if and only if its every block is either a simple cycle or a single edge. A cactus is a tree if and only if its each block is an edge. An edge of a cactus is a cycle edge if it is in some cycle, and tree edge, otherwise. A cactus is said to be a bundle if there is a single common vertex on all of its cycles. Let B₁(n) be the bundle of order n + obtained from a star K_1,n of the same order with central vertex V₀ by adding the edges v_iv_i+1, where i ∈ {1, 3, 5, …, n – 1} and n = 0 mod(2). Thus, the central vertex v₀ of the bundle B₁(n) has degree n and each remaining vertex is of degree 2. Similarly, let B₂(n) be a bundle obtained from K_1,n by adding the edges v_iv_i+1, where i ∈ {1, 3, 5, …, n – 2}, n = 1 mod(2) and |V(B₂(n))| = n + 1. Thus, for the vertices of B₂(n), d(v₀) = n, d(v_n) = 1 and d(v_i) = 2, where 1 ≤ i ≤ n – 1.

We define some particular cacti which are obtained from the aforesaid bundles.

Definition 2.1

Assume that p, q = 0 mod(2) are positive integers. Let B₁(p) and B₁(q) be two bundles. The cactus graph C₁(p, q) is constructed by the join of a vertex of B₁(p) with a vertex B₁(q), where both the vertices are of degree 2. Thus, V(C₁(p, q)) = { v1j : 1 ≤ j ≤ p – 2} ∪ {v_j : 2 ≤ j ≤ 7} ∪ { v8j : 1 ≤ j ≤ q – 2} and E(C₁(p, q)) = {v₂ v1j : 1 ≤ j ≤ p – 2} ∪ {v₂v₃, v₂v₄} ∪ {v1jv1j+1:j=1,3,5,...,p−3} ∪ {v_jv_j+1 : 3 ≤ j ≤ 5} ∪ {v₇v₅, v₇v₆} ∪ {v7v8j:1≤j≤q−2}∪{v8jv8j+1:j=1,3,5,...,q−3} with |C₁(p, q)| = 2 + p + q = n.

Assume that p, q ≡ 1 mod(2) and p, q ≥ 3. If a vertex of the bundle B₂(p) is joined with a vertex of the bundle B₂(q) then we obtain the cactus graph C1′ (p, q), where both the chosen vertices are pendent and n = 2 + p + q = | C1′ (p, q)|. For p ≥ 3, q ≥ 2, p ≡ 1 mod(2) and q ≡ 0 mod(2), if we join a vertex of the bundle B₂(p) to a vertex of the bundle B₁(q) then we obtain a cactus graph C₂(p, q), where the chosen vertices are of degree 1 and 2 respectively and n = 2 + p + q = |C₂(p, q)|. Similarly, if we assume p ≥ 2, q ≥ 3, p = 0 mod(2) and q = 1 mod(2), and choose two vertices of degree 2 and 1 in B₁(p) and B₂(q) respectively. On joining these chosen vertices by an edge, we obtain the cactus graph C2′ (p, q) with n = 2 + p + q = | C1′ (p, q)|.

We note that C₁(p, q) ≅ C₁(q, p) and C1′ (p, q) ≅ C1′ (q, p) as p and q both are even in C₁(p, q) and odd in C1′ (p, q). Moreover, as p is odd and q is even in C₂(p, q), and p is even and q is odd in C2′ (p, q) therefore C₂(p, q) ≇ C₂(q, p), C2′ (p, q) ≇ C2′ (q, p) and C₂(q, p) ≅ C2′ (p, q). The cacti C₁(p, q) and C1′ (p, q) are presented in Figure 1((a) and (b)) and the cacti C₂(p, q) and C2′ (p, q) are presented in Figure 2((a) and (b)).

$Figure 1 (a)C1(p, q) and (b) C1′ $\begin{array}{} C^{'}_1 \end{array}$(p, q).$

Figure 1

(a)C₁(p, q) and (b) C1′ (p, q).

$Figure 2 (a)C2(p, q) and (b) C2′ $\begin{array}{} C^{'}_2 \end{array}$(p, q).$

Figure 2

(a)C₂(p, q) and (b) C2′ (p, q).

Let Ω_1,n be the class of cacti other than stars such that each block of a cactus is an edge and Ω_2,n be a class of cacti other than bundles such that at least one block of each cactus is an edge and at least one block is a cycle. Let Ω_n be a class of cacti other than stars and bundles such that either all the blocks of a cactus are edges or a cactus has at least one block which is a cycle and at least one block which is an edge, i.e. Ω_n = Ω_1,n ∪ Ω_2,n. Thus, we obatain Ωnc = {Γ^c : Γ^c is connected, |Γ^c| = n ∧ Γ ∈ Ω_n}. By interlacing theorem, λ_min(Γ) ≤ –1 if Γ contains at least one edge. Moreover, equality holds if Γ is a complete graph. Another way to achieve this equality is if Γ = ∪_i G_i, where all G_i are complete graphs and at least one G_i is non-trivial. Thus, for Γ ∈ Ω_n, λ_min(Γ^c) < –1.

If ϕ^′ : V(Γ) → {X_i : 1 ≤ i ≤ n} is a 1-1 map such that ϕ^′(u_i) = X_i for each u_i ∈ V(Γ) then it is said to be defined on the graph Γ. The eigenvector X of A(Γ) is naturally defined on V(Γ). Thus, we have

XTAX=2∑uv∈E(Γ)XuXv. (2.1)

The eigenequation for each v ∈ V(Γ) is

λXv=∑u∈NΓ(v)Xu, (2.2)

where all adjacent to v are in N_Γ(v). If X ∈ Rⁿ is an arbitrary unit vector, we have

λmin(Γ)≤XTA(Γ)X, (2.3)

where equality holds iff X is a FEv. If Γ^c is complement of Γ, then A(Γ^c) = J – I – A(Γ) with J and I as all-ones and identity matrix respectively. Thus, for X ∈ Rⁿ

XTA(Γc)X=XT(J−I)X−XTA(Γ)X. (2.4)

Let Y₁ be FEv of C₁(p, q)^c which is defined on it. By (2.2), the vertices v1j for 1 ≤ j ≤ p – 2, v₂, v₃, v₄, v₅, v₆, v₇ and v8j for 1 ≤ j ≤ q – 2 having values in Y₁, say X_j for 1 ≤ j ≤ 8 respectively. If λ_min (C₁(p, q)^c) = λ₁ then

λ1X1=(p−4)X1+X3+X4+X5+X6+X7+(q−2)X8,λ1X2=X5+X6+X7+(q−2)X8,λ1X3=(p−2)X1+X5+X6+X7+(q−2)X8,λ1X4=(p−2)X1+X6+X7+(q−2)X8,λ1X5=(p−2)X1+X2+X3+(q−2)X8,λ1X6=(p−2)X1+X2+X3+X4+(q−2)X8,λ1X7=(p−2)X1+X2+X3+X4,λ1X8=(p−2)X1+X2+X3+X4+X5+X6+(q−4)X8. (2.5)

Take Y₁ = (X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈)^T then the matrix equation is (A – λ₁I)Y₁ = 0 and

f1(λ,p,q)=det(A−λI)=(4+pq−2p−2q)+(16−4p−4q)λ+(−4−7pq+12p+12q)λ2+(−48+2pq+14p+14q)λ3+(−20+7pq−9p−9q)λ4+(24+2pq−16p−16q)λ5+(24−7p−7q)λ6+(8−p−q)λ7+λ8... (2.6)

with least root λ₁.

Let Y1′ be FEv of C1′ (p, q)^c. By (2.2), the vertices v1j for 1 ≤ j ≤ p – 1, v₂, v₃, v₄, v₅ and v6j for 1 ≤ j ≤ q – 1 having values in Y1′ , say X_j for 1 ≤ j ≤ 6 respectively. If λ_min ( C1′ (p, q)^c) = λ1′ then

λ1′X1=(p−3)X1+X3+X4+X5+(q−1)X6,λ1′X2=X4+X5+(q−1)X6,λ1′X3=(p−1)X1+X5+(q−1)X6,λ1′X4=(p−1)X1+X2+(q−1)X6,λ1′X5=(p−1)X1+X2+X3,λ1′X6=(p−1)X1+X2+X3+X4+(q−1)X6. (2.7)

Take Y1′ = (X₁, X₂, X₃, X₄, X₅, X₆)^T then the matrix equation is (A – λ1′ I) Y1′ = 0 and

f1′(λ,p,q)=det(A−λI)=(8−2p−2q)+(−8−2pq+7p+7q)λ+(−18+3pq+2p+2q)λ2+(2pq−7p−7q)λ3+(11−5p−5q)λ4+(6−p−q)λ5+λ6... (2.8)

with least root λ1′ .

Let Y₂ be FEv of C₂(p, q)^c. By (2.2), the vertices v1j for 1 ≤ i ≤ p – 1, v₂, v₃, v₄, v₅, v₆, and v7j for 1 ≤ j ≤ q – 2 having values in Y₂, say X_j for 1 ≤ j ≤ 7 respectively. If λ_min (C₂(p, q)^c) = λ₂ then

λ2X1=(p−3)X1+X3+X4+X5+X6+(q−2)X7,λ2X2=X4+X5+X6+(q−2)X7,λ2X3=(p−1)X1+X5+X6+(q−2)X7,λ2X4=(p−1)X1+X2+(q−2)X7,λ2X5=(p−1)X1+X2+X3+(q−2)X7,λ2X6=(p−1)X1+X2+X3,λ2X7=(p−1)X1+X2+X3+X4+X5+(q−4)X7. (2.9)

Take Y₂ = (X₁, X₂, X₃, X₄, X₅, X₆, X₇)^T then the matrix equation is (A – λ₂ I)Y₂ = 0 and

f2(λ,p,q)=det(A−λI)=(−6−pq+2p+3q)+(−8−2p−2q+3pq)λ+(26+pq−10p−13q)λ2+(23−5pq+3p+q)λ3+(−9−2pq+11p+11q)λ4+(−17+6p+6q)λ5+(−7+p+q)λ6−λ7... (2.10)

with least root λ₂.

Let Y2′ be the FEv of C2′ (p, q)^c. By (2.2), the vertices v1j for 1 ≤ j ≤ p – 2, v₂, v₃, v₄, v₅, v₆, and v7j for 1 ≤ j ≤ q – 1 having values in Y2′ , say X_j for 1 ≤ j ≤ 7 respectively. If λ_min ( C2′ (p, q)^c) = λ₂ then

λ2′X1=(p−4)X1+X3+X4+X5+X6+(q−1)X7,λ2′X2=X5+X6+(q−1)X7,λ2′X3=(p−2)X1+X5+X6+(q−1)X7,λ2′X4=(p−2)X1+X6+(q−1)X7,λ2′X5=(p−2)X1+X2+X3+(q−1)X7,λ2′X6=(p−2)X1+X2+X3+X4,λ2′X7=(p−2)X1+X2+X3+X4+X5+(q−3)X7. (2.11)

Take Y2′ = (X₁, X₂, X₃, X₄, X₅, X₆, X₇)^T, the matrix equation is (A – λ2′ I) Y2′ = 0 and

f2′(λ,p,q)=det(A−λI)=(−6−pq+2p+3q)+(−8−2p−2q+3pq)λ+(26+pq−13p−10q)λ2+(23−5pq+p+3q)λ3+(−9−2pq+11p+11q)λ4+(−17+6p+6q)λ5+(−7+p+q)λ6−λ7... (2.12)

with least root λ2′ .

3 Minimizing graphs

Now, we present some important lemmas of the minimizing graph which are frequently used in next section. The classes of cacti which have graphs of even order are discussed from Lemma 3.1 to Lemma 3.6. Moreover, the cacti of odd order are studied from Lemma 3.7 to Lemma 3.10.

Firstly, we discuss the classes of cacti which have graphs of even order.

Lemma 3.1

Suppose that p, q ≥ 4, n ≥ 12 are integers with p, q, n ≡ 0(mod 2). If p > q + 2, then

λmin(C1(p−2,q+2)c)<λmin(C1(p,q)c),

where p + q + 2 = n = |V(C₁(p – 2, q + 2)^c)| = |V(C₁(p, q)^c)|.

Proof

From equation (2.5), we have f₁(–3, p, q) = 325 – 17(p + q) – 35pq. Since for p, q ≥ 4 f₁(–3, p, q) < 0. Therefore, least root of f₁(λ, p, q) is λ₁ < –3. Moreover, f₁(λ, p – 2, q + 2) = (pq – 4q) + (16 – 4p – 4q)λ + (24 – 2p + 26q – 7pq)λ² + (–56 + 18p + 10q + 2pq)λ³ + (–48 + 5p – 23q + 7pq)λ⁴ + (16 – 12p – 20q + 2pq)λ⁵ + (24 – 7p – 7q)λ⁶ + (8 – p – q)λ⁷ + λ⁸, and

f1(λ,p,q)−f1(λ,p−2,q+2)=−2(p−q−2)(2λ5+7λ4+2λ3−7λ2+1)=−2(p−q−2)(λ−12)(λ+3+52)(λ+3−52)(λ+1+52)(λ+1−52).

As p is greater than q + 2 and λ is less than –3 therefore f₁(λ, p, q)- f₁(λ, p – 2, q + 2) > 0. Also, f₁(–3, p – 2, q + 2) < 0 which implies that λ_min(C₁(p – 2, q + 2)^c) < λ_min(C₁(p, q)^c).

Corollary 3.2

Suppose that p, q ≥ 4, n ≥ 12 are integers with p, q, n ≡ 0(mod 2). If q > p + 2, then λ_min(C₁(p + 2, q – 2)^c) < λ_min(C₁(p, q)^c), where p + q + 2 = n.

Proof

Since, C₁(p, q)^c ≅ C₁(q, p)^c, therefore proof is same as of Lemma 3.1.

Lemma 3.3

λmin(C1(p,q)c)≥λmin(C1(n−22,n−22)c)aifan≡2(moda4);λmin(C1(n2,n−42)c)aaaifan≡0(moda4);

where equality holds iff p = n−22 = q with n ≥ 14 and p = n2 and q = n−42 with n ≥ 12.

Proof

When n ≡ 2(mod 4), then for p = n−22 = q, the equation (2.5) becomes f1(−3,n−22,n−22) = –(n – 7.2)(n + 5.1428). For n ≥ 14, we have f1(−3,n−22,n−22) < 0. (b) When n ≡ 0(mod 4), then for p = n2 and q = n−42 , the equation (2.5) becomes f1(−3,n2,n−42) = –(n – 7.5159)(n + 5.4588). For n ≥ 12, we have f1(−3,n2,n−42) < 0. Thus, from both the cases least root of f₁(λ, p, q) is λ₁ < –3.

Now, by Lemma 3.1, if q + 2 < p and λ < –3, then λ_min(C₁(p – 2, q + 2)^c) < λ_min(C₁(p, q)^c) and by Corollary 3.1, if q > p + 2 and λ < –3, then λ_min(C₁(p + 2, q – 2)^c) < λ_min(C₁(p, q)^c).

Consequently, for n ≥ 14 and n ≡ 2(mod 4), we have λmin(C1(n−22,n−22)c)≤λmin(C1(p,q)c) with equality iff p = n−22 = q, and (b) for n ≥ 12 and n ≡ 0(mod 4), we have λmin(C1(n2,n−42)c)≤λmin(C1(p,q)c) with equality iff p = n2 and q = n−42 .

Lemma 3.4

Suppose that p ≥ 5, q ≥ 3, n ≥ 12 are integers with p, q ≡ 1(mod 2) and n ≡ 0(mod 2). If p > q + 2, then

λmin(C1′(p−2,q+2)c)<λmin(C1′(p,q)c),

where p + q + 2 = n is cardinality of both the cacti.

Proof

By (2.8), f1′ (–3, p, q) = 32 + (p + q) – 21(pq – p – q). Since, for p ≥ 5 and q ≥ 3, f1′ (–3, p, q) < 0. Therefore, least root f1′ (λ, p, q) is λ₁ < –3. Also,

f1′(λ,p−2,q+2)=(8−2p−2q)+(−2pq+3p+11q)λ+(−30+3pq+8p−4q)λ2+(2pq−3p−11q−8)λ3+(11−5p−5q)λ4+(6−p−q)λ5+λ6

f1′(λ,p,q)−f1′(λ,p−2,q+2)=(8−4p+4q)λ3+(12−6p+6q)λ2+(−8+4p−4q)=−2λ(λ+2)(λ−0.5).

Since p is greater than q + 2 and λ is less than –3, f1′ (λ, p, q) – f1′ (λ, p – 2, q + 2) > 0. Also, f1′ (–3, p – 2, q + 2) < 0 which implies that λ_min( C1′ (p – 2, q + 2)^c) < λ_min( C1′ (p, q)^c).

Corollary 3.5

Suppose that p ≥ 3, q ≥ 5, n ≥ 12 are integers with p, q ≡ 1(mod 2) and n ≡ 0(mod 2). If q > p + 2, then λ_min( C1′ (p + 2, q – 2)^c) < λ_min( C1′ (p, q)^c), where p + q + 2 = n.

Proof

Since, C1′ (p, q)^c ≅ C1′ (q, p)^c, therefore proof is same as of Lemma 3.4.

Lemma 3.6

λmin(C1′(p,q)c)≥λmin(C1′(n2,n−42)c)aaaifan≡2(moda4);λmin(C1(n2,n−42)c)aaaifan≡0(moda4);

where equality holds iff p = n2 and q = n−42 with n ≥ 14, and p = n−22 = q with n ≥ 12.

Proof

When n ≡ 2(mod 4), then for p = n2 and q = n−42 the equation (2.8) becomes f1′(−3,n2,n−22)=−14 (n – 6.4842)(n + 0.2937). For n ≥ 14, we have f1(−3,n2,n−22) < 0. (b) When n ≡ 0(mod 4), then for p = n−22 = q the equation (2.8) becomes f1′(−3,n−22,n−22)=−14 (n – 7.3333)(n – 0.8571). For n ≥ 12, we have f1′(−3,n−22,n−22) < 0. Thus, from both the cases least root of f1′ (λ, p, q) is λ1′ < –3.

Now, by Lemma 3.4, for p > q + 2 and λ < –3, λ_min( C1′ (p – 2, q + 2)^c) < λ_min( C1′ (p, q)^c) and by Corollary 3.5, if q > p + 2 and λ < –3, then λ_min( C1′ (p + 2, q – 2)^c) < λ_min( C1′ (p, q)^c).

Consequently, for n ≥ 14 and n ≡ 2(mod 4), we have λmin(C1′(n2,n−42)c)≤λmin(C1′(p,q)c) with equality iff p = n2 and q = n−42 , and (b) for n ≥ 12 and n ≡ 0(mod 4), we have λmin(C1′(n−22,n−22)c)≤λmin(C1′(p,q)c) with equality iff p = n−22 = q.

Now, we discuss the classes of graphs having graphs of odd order.

Lemma 3.7

Suppose that p ≥ 5, q ≥ 2 and n ≥ 13 are integers with p, n ≡ 1(mod 2) and q ≡ 0(mod 2). If p > q + 3, then

λmin(C2(p−2,q+2)c)<λmin(C2(p,q)c),

where p + q + 2 = n = |V(C₂(p – 2, q + 2)^c)| = |V(C₂(p, q)^c)|.

Proof

From equation (2.10), we have f₂(–3, p, q) = 117 – (p + q) – 28q(p – 1). Since, for p ≥ 5 and q ≥ 2 f₂(–3, p, q) < 0. Therefore, least root of f₂(λ, p, q) is λ₂ < –3. Also, f₂(λ, p – 2, q + 2) = +(5q – pq) + (–20 + 4p – 8q + 3pq)λ + (16 – 8p – 15q + pq)λ² + (39 – 7p + 11q – 5pq)λ³ + (–1 + 7p + 15q – 2pq) λ⁴ + (–17 + 6p + 6q)λ⁵ + (–7 + p + q)λ⁶ – λ⁷, and

f2(λ,p,q)−f2(λ,p−2,q+2)=4(λ+1)(λ−12)(λ−α1+α2α3)(λ−α1−α2α3),

where α1=3−2(p−q),α2=8(p−q)2−32(p−q)+33 and α₃ = 2((p – q) – 2).

If p – q → 4, then α1+α2α3 → 0 and α1−α2α3 → –3. If p – q → +∞, then α1+α2α3 → –1 + 2 and α1−α2α3 → –1 – 2 . This shows that for p – q ≥ 4, α1+α2α3 ∈ [0, 0.4142[ and α1−α2α3 ∈ [–3, –2.4142[. Thus, for p > q + 3 and λ < –3, we have f₂(λ, p, q) – f₂(λ, p – 2, q + 2) > 0. Also, f₂(–3, p – 2, q + 2) < 0 which implies that λ_min(C₂(p – 2, q + 2)^c) < λ_min(C₂(p, q)^c).

Lemma 3.8

Suppose that p ≥ 4, q ≥ 3 and n ≥ 13 are integers with q, n ≡ 1(mod 2) and p ≡ 0(mod 2). If p > q + 3, then

λmin(C2′(p−2,q+2)c)<λmin(C2′(p,q)c),

where p + q + 2 = n.

Proof

From equation (2.12), we have f2′ (–3, p, q) = 117 – (p + q) – 28p(q – 1). Since, for p ≥ 4 and q ≥ 3, f2′ (–3, p, q) < 0. Therefore, least root of f2′ (λ, p, q) is λ2′ < –3. Also,

f2′(λ,p−2,q+2)=(−4+p+4q−pq)+(−20+4p−8q+3pq)λ+(28−11p−12q+pq)λ2+(47−9p+13q−5pq)λ3+(−1+7p+15q−2pq)λ4+(−17+6p+6q)λ5+(−7+p+q)λ6−λ7,

and

f2′(λ,p,q)−f2′(λ,p−2,q+2)=2(λ+1)(λ−0.5)(λ−0.1861)(λ+2.6861).

Thus, for p > q and λ < –3, we have f2′ (λ, p, q) – f2′ (λ, p – 2, q + 2) > 0. Also, f2′ (–3, p – 2, q + 2) < 0 which implies that λ_min( C2′ (p – 2, q + 2)^c) < λ_min( C2′ (p, q)^c).

Corollary 3.9

Suppose that p ≥ 3, q ≥ 4 and n ≥ 13 are integers with p, n ≡ 1(mod 2) and q ≡ 0(mod 2). If q > p + 3, then λ_min(C₂(p + 2, q – 2)^c) < λ_min(C₂(p, q)^c), where p + q + 2 = n.

Proof

Using Lemma 3.8, if r > s + 3 and r + s + 2 = n then

λmin(C2′(r−2,s+2)c)<λmin(C2′(r,s)c).

Therefore, for n ≡ 1(mod 4), we have

λmin(C2′(n−7,5)c)<λmin(C2′(n−5,3)c),λmin(C2′(n−9,7)c)<λmin(C2′(n−7,5)c),...,λmin(C2′(n−52,n+12)c)<λmin(C2′(n−12,n−32)c).

Similarly for n ≡ 3(mod 4), we have

λmin(C2′(n−7,5)c)<λmin(C2′(n−5,3)c),λmin(C2′(n−9,7)c)<λmin(C2′(n−7,5)c),...,λmin(C2′(n−32,n−12)c)<λmin(C2′(n+12,n−52)c).

Now, by definition, for n ≡ 1(mod 4), C2′ (n – 5, 3) ≅ C₂(3, n – 5), C2′ (n – 7, 5) ≅ C₂(5, n – 7), …, C2′(n−52,n+12)≅C2(n+12,n−52) and for n ≡ 3(mod 4)’ C2′ (n – 5, 3) ≅ C₂(3, n – 5), C2′ (n – 7, 5) ≅ C₂(5, n – 7), …, C2′(n−32,n−12)≅C2(n−12,n−32).

Consequently, λ_min(C₂(p + 2, q – 2)^c) < λ_min(C₂(p, q)^c) for q > p, which complete the proof.

Lemma 3.10

Suppose that p ≥ 3 and q ≥ 2 are integers with p ≡ 1(mod 2) and q ≡ 0(mod 2), where p + q + 2 = n is of the cacti. Then

λmin(C2(p,q)c)≥λmin(C2(n+12,n−52)c)aaaifan≡1(moda4);λmin(C2(n−12,n−32)c)aaaifan≡3(moda4);

where equality holds iff p = n+12 and q = n−52 with n ≥ 13, and p = n−12 and q = n−32 with n ≥ 15.

Proof

When n ≡ 1(mod 4), then for p = n+12 and q = n−52 , the equations (2.10) becomes f2(−3,n+12,n−52) = –(n – 7.4647)(n + 1.6075). Thus, for n ≥ 13, we have f2(−3,n+12,n−52) < 0. (b) When n ≡ 3(mod 4), then for p = n−12 and q = n−32 , the equation (2.10) becomes, f2(−3,n−12,n−32) = –(n – 5.7569)(n – 0.1001). So, for n ≥ 15, we have f2(−3,n−12,n−32) < 0. Thus, from both the cases least root of f₂(λ, p, q) is λ₂ < –3.

Now, by Lemma 3.7, if p is greater than q + 3 and λ is less than –3, then λ_min(C₂(p – 2, q + 2)^c) < λ_min(C₂(p, q)^c) and by Corollary 3.9, if q > p, q – p ≠ 2 and λ < –3, then λ_min(C₂(p + 2, q – 2)^c) < λ_min(C₂(p, q)^c).

Consequently, for n ≥ 13 and n ≡ 1(mod 4), we have λmin(C2(n+12,n−52)c)≤λmin(C2(p,q)c) with equality if p = n+12 and p = n−52 , and (b) for n ≥ 15 and n ≡ 3(mod 4), we have λmin(C2(n−12,n−32)c)≤λmin(C2(p,q)c) with equality iff p = n−12 and q = n−32 . This complete the proof.

4 Characterization

This section includes the main results in which minimizing graphs are characterized in the family of connected graphs with the condition that the complement of each graph is a cactus such that either its each block is only an edge or it has at least one block which is an edge and at least one block which is a cycle. In Lemma 4.1 and 4.2, the basic results are developed which are used in the main results. In Lemma 4.3 and Lemma 4.4, minimizing graphs are characterized in Ω1,nc and Ω2,nc respectively. Finally, in Theorem 4.5 minimizing graphs are characterized in Ωnc=Ω1,nc∪Ω2,nc.

Lemma 4.1

Let n ≥ 12 and X = (X₁, X₂, X₃, …, X_n)^T be a real vector defined on C ∈ Ω_n such that |X₁| ≥ |X₂| ≥ |X₃| ≥ ... ≥ |X_n| and all X_i are either non-negative or non-positive. Then

∑uv∈E(C)XuXv≤∑uv∈E(B1(n−1))XuXvaifan≡1(mod2);∑uv∈E(B2(n−1))XuXvaifan≡0(mod2);

where equality holds if C ≅ B₁(n – 1) and C ≅ B₂(n – 1) respectively.

Proof

Suppose that X is non-negative and discuss the following two cases:

Suppose that C ∈ Ω_n is a cactus graph such that its each block is an edge i.e. C ∈ Ω_1,n. Let X₁ be the value of the vertex v ∈ C assigned by X. As n ≥ 10 and is not a bundle, therefore there exist a vertex in C say u which is not adjacent to v. Thus, a vertex w adjacent to u (w ∼ u) exists on a path from v to u as C is connected. A new cactus C̃ having each block as an edge can be found on the deletion of the edge wu and addition of vu in C. We find a star K_1,n–1 with center v by repeating the same process for the non-neighbor of v in the cactus C̃ and so on. Thus, we have

∑uv∈E(C)XuXv≤∑uv∈E(C~)XuXv≤∑i=2nX1Xi=∑uv∈E(K1,n−1)XuXv (4.1)
Suppose that C ∈ Ω_n is a cactus graph such that it has at least one block which is an edge and at least one block which is a cycle i.e. C ∈ Ω_2,n. Assume that k is number of cycles in C, then 1≤k≤⌈n−22⌉, where n is order of C. Now, we delete an edge in each cycle under the conditions, (i) no two deleted edges of any two cycles have common vertex and (ii) the deleted edge in a cycle which has the vertex with label X₁ is not incident on this vertex. Thus, we have a cactus graph such that its each block is an edge, say C^′ i.e. C^′ ∈ Ω_1,n. Then by Case 1, we have ∑uv∈E(C′)XuXv≤∑uv∈E(K1,n−1)XuXv. Now, we add the deleted edges, then

∑uv∈E(C)XuXvaa=∑uv∈E(C′)XuXv+(Xvr1Xvr1+1+Xvr2Xvr2+1+,...,+XvrkXvrk+1)⏟katerms≤∑uv∈E(K1,n−1)XuXv+(Xvr1Xvr1+1+Xvr2Xvr2+1+,...,+XvrkXvrk+1),

where v_r, v_r+1, v_s, v_s+1, …, v_t, v_t+1 are 2k distinct vertices of C. Moreover, the inequality in (4.1) does not disturb, if we add k terms (X_{v_r₁} X_{v_r₁+1} + X_{v_r₂} X_{v_r₂+1} +, …, + X_{v_{r_k}} X_{v_{r_k+1}}) in its right hand side. Consequently, from both the cases, we have

∑uv∈E(C)XuXv≤∑uv∈E(K1,n−1)XuXv+(Xvr1Xvr1+1+Xvr2Xvr2+1+,...,+XvrkXvrk+1). (4.2)

Assume n ≡ 1(mod 2). Since, in K_1,n–1, one vertex, say v₁ with d(v₁) = n – 1 has value X₁ and remaining n – 1 pendent vertices have values X_i for 2 ≤ i ≤ n. Pairing the vertices of degree 1 and joining each pair by an edge, we have n−12 edges v_iv_i+1, where i ∈ I = {2, 4, …, n – 1}. Thus, (X_{v_r} X_{v_r+1} + X_{v_s} X_{v_s+1} +, …, + X_{v_t} X_{v_t+1}) ≤ ∑i∈I X_iX_i+1. Since, ∑uv∈E(K1,n−1) X_uX_v + ∑i∈I X_iX_i+1 = ∑uv∈E(B1(n−1)) X_uX_v. Consequently (4.2) becomes, ∑uv∈E(C) X_uX_v ≤ ∑uv∈E(B1(n−1)) X_uX_v, where equality holds if C has one vertex of degree n – 1 and remaining n – 1 vertices are of degree two, i.e. C ≅ B₁(n – 1).
Assume n ≡ 0(mod 2). Since, in K_1,n–1, one vertex, say v₁ with d(v₁) = n – 1 has value X₁ and remaining n – 1 pendent vertices have values X_i for 2 ≤ i ≤ n. Pairing the n – 2 vertices of degree 1 and joining each pair by an edge, we have n−22 edges v_iv_i+1, where i ∈ I = {2, 4, …, n – 2}. Thus, (X_{v_r₁} X_{v_r₁+1} + X_{v_r₂} X_{v_r₂+1} +, …, + X_{v_{r_k}} X_{v_{r_k+1}}) ≤ ∑i∈I X_iX_i+1. Since, ∑uv∈E(K1,n−1) X_uX_v + ∑i∈I X_iX_i+1 = ∑uv∈E(B2(n−1)) X_uX_v.

So, (4.2) takes form ∑uv∈E(C) X_uX_v ≤ ∑uv∈E(B2(n−1)) X_uX_v, where equality holds if C has one pendent vertex, one of degree n – 1 and n – 2 with degree 2, i.e. C = B₂(n – 1).

Lemma 4.2

For n ≥ 12 and C^c ∈ Ωnc , the first eigenvector X of C^c has at least 2 negative and 2 positive entries.

Proof

Assume that there is a unique vertex v ∈ C^c having positive value labeled by X. The degree of v in C^c is non-zero, i.e d_C^c(v) ≠ 0. As, if d_C^c(v) = 0, then C is a bundle, which is a contradiction to the construction of Ωnc . Consequently, 1 ≤ d_C^c(v) ≤ n – 1. Let u be another vertex in C^c. Since all the vertices are supposed to have negative values except v, we claim u ∼ v, otherwise (2.2) does not holds for u ∈ C^c as λ X_u > 0 and ∑w∈NG(u) X_w < 0 ⇒ λX_u ≠ ∑w∈NG(u) X_w. Consequently, d_C^c(v) = n – 1 for each v ∈ C^c. It shows that C is disconnected, which is a contradiction to the construction of Ωnc . Similarly, we can prove, if v ∈ C^c is a unique vertex with negative value assigned by X.

Lemma 4.3

Suppose that C is a cactus graph such that C^c ∈ Ω2,nc and |V(C)| = n = p + q + 2 ≥ 12.

If n ≡ 0(mod 2), then either λ_min(C₁(p, q)^c) ≤ λ_min(C^c), or λ_min( C1′ (p, q)^c) ≤ λ_min(C^c), where equalities hold if C ≅ C₁(p, q) or C ≅ C1′ (p, q), respectively.
If n ≡ 1(mod 2), then λ_min(C₂(p, q)^c) ≤ λ_min(C^c) with equality if C ≅ C₂(p, q).

Proof

Assume X is a unit first eigenvector of C^c. Define V₊ = {v : X_v ≥ 0, v ∈ V(C^c)} and V_– = {v : X_v < 0, v ∈ V(C^c)} such that |V₊|, |V_–| ≥ 2 by Lemma 4.2. Suppose that the subgraphs C₊ and C_– of C are induced by the vertex sets V₊ and V_– respectively and E^′ ≠ Φ is subset of E(C) with one end in C₊ and other in C_–. Thus, we have

∑uv∈E(C)XuXv=∑uv∈E(C+)XuXv+∑uv∈E(C−)XuXv+∑uv∈E′XuXv. (4.3)

Assume that n ≡ 0(mod 2), where n = p + q + 2. Let C̄ be a graph obtained from C by some possible addition or deletion of edges in C₊ and C_– such that the subgraph C̄₊ and C̄_– of C̄ induced by C₊ and C_– are cactus graphs satisfying one of the following possibilities, (i) each block of one of the subgraphs C̄₊ and C̄_– is an edge and other has at least one block which is a cycle and at least one block which is an edge, (ii) all the blocks of both the subgraphs C̄₊ and C̄_– are cycles, (iii) all the blocks of one subgraph are edges and of other are cycles, (iv) each block of one of the subgraphs C̄₊ and C̄_– is a cycle and other has at least one block which is a cycle and at least one block which is an edge, and (v) both the subgraphs C̄₊ and C̄_– have at least one block which is a cycle and at least one block which is an edge.

For (i), suppose C̄₊ is a cactus such that its each block is an edge, otherwise we take –X as a first eigenvector. Let u^′ be a vertex of C̄₊ with maximum modulus among all the vertices, then by discussion of equation (4.1) in Lemma 4.1, we obtain a cactus with each block as an edge which is infect a star K_1,p. Similarly, suppose that v^′ is a vertex with maximum modulus among all the vertices of C̄_–. Firstly, we delete an edge in each block such that no two deleted edges of any two blocks have a common vertex in C̄_– and the deleted edge in a block which has v^′ is not incident on this vertex. Thus, we obtain a subgraph of C̄_– such that its each block is an edge. Then by the same discussion as of equation (4.2) in Lemma 4.1, we obtain a cactus with at least one block as a cycle and at least one block as an edge which is infect a star K_1,q with edges among the pendent vertices having different end points.

Since n ≡ 0(mod 2) and n = p + q + 2, where n = |V₊ ∪ V_–| = |C̄₊ ∪ C̄_–|, p + 1 = |V₊| = |C̄₊| and q + 1 = |V_–| = |C̄_–|. Therefore, either both p and q are even or odd.

Suppose p and q both are even. By pairing the pendent vertices of the star K_1,p which is obtained from C̄₊ and joining them by edges, we have a bundle B₁(p) with center u^′ having maximum modulus value, where p + 1 = |V₊| ≥ 3 is odd. Similarly, pair the remaining possible pendent vertices of the subgraph obtained from C̄_– and join them by edges. Thus, we obtain a bundle B₁(q) with center v^′ having maximum modulus value, where q + 1 = |V₊| ≥ 3 is odd. Thus, by Lemma 4.1 (a), we have

∑uv∈E(C+)XuXv≤∑uv∈E(C¯+)XuXv≤∑uv∈E(B1(p))XuXv (4.4)

and

∑uv∈E(C−)XuXv≤∑uv∈E(C¯−)XuXv≤∑uv∈E(B1(q))XuXv. (4.5)

Suppose p and q both are odd. By pairing the pendent vertices of the star which is obtained from C̄₊ and joining them by edges, we have a bundle B₂(p) with center u^′ having maximum modulus value, where p + 1 = |V₊| ≥ 4 is even. Similarly, pair the remaining possible pendent vertices of the subgraph obtained from C̄_– and join them by edges. Thus, we obtain a bundle B₂(q) with center v^′ having maximum modulus value, where q + 1 = |V₊| ≥ 4 is even. Thus, by Lemma 4.1 (b), we have

∑uv∈E(C+)XuXv≤∑uv∈E(C¯+)XuXv≤∑uv∈E(B2(p))XuXv, (4.6)

∑uv∈E(C−)XuXv≤∑uv∈E(C¯−)XuXv≤∑uv∈E(B2(q))XuXv. (4.7)

Assume that u^″ ∈ C̄₊ and v^″ ∈ C̄_– have minimum modulus values. Then

∑uv∈E′XuXv≤Xu″Xv″. (4.8)

Using (4.4), (4.5), (4.8) and (4.6), (4.7), (4.8) in (4.3) respectively, we have

∑uv∈E(C)XuXv≤∑uv∈E(B1(p))XuXv+∑uv∈E(B1(q))XuXv+Xu″Xv″, (4.9)

∑uv∈E(C)XuXv≤∑uv∈E(B2(p))XuXv+∑uv∈E(B2(q))XuXv+Xu″Xv″. (4.10)

Since p ≥ q ≥ 2. Therefore, if we take u^″ ∈ B₁(p), v^″ ∈ B₁(q) of degree 2 and u^″ ∈ B₂(p), v^″ ∈ B₂(q) of degree 1. Then (4.9) and (4.10) becomes,

∑uv∈E(C)XuXv≤∑uv∈E(C1(p,q))XuXv, (4.11)

∑uv∈E(C)XuXv≤∑uv∈E(C1′(p,q))XuXv. (4.12)

Now by the equations (2.1)-(2.4) and (4.11), we have λ_min(C^c) = X^TA(C^c)X = X^T(J – I – A(C))X = X^T(J – I)X – X^TA(C)X ≥ X^T(J – I)X – X^TA(C₁(p, q))X = X^TA(C₁(p, q)^c)X ≥ λ_min(C₁(p, q)^c) ⇒ λ_min(C₁(p, q)^c) ≤ λ_min(C^c). Similarly by equations (2.1)-(2.4) and (4.12), we have λ_min( C1′ (p, q)^c) ≤ λ_min(C^c).

Thus, for n ≡ 0(mod 2), either λ_min(C₁(p, q)^c) ≤ λ_min(C^c) or λ_min( C1′ (p, q)^c) ≤ λ_min(C^c), where p + q + 2 = n. On the same way, it can be prove that the results are also true for all the possibilities (ii)-(v).
Assume that n ≡ 1(mod 2), where n = p + q + 2. Let C̄ be a graph obtained from C with subgraph C̄₊ and C̄_– induced from C₊ and C_– are cactus graphs satisfying any one of the possibilities which are stated in (a). For (i), we proceed same as in (a) and find the cactus graphs which are infect stars with some possible edges among the pendent vertices having different end points from both the subgraphs C̄₊ and C̄_– after the deletion and addition of some edges. Since n ≡ 1(mod 2) and n = p + q + 2, where n = |V₊ ∪ V_–| = |C̄₊ ∪ C̄_–|, p + 1 = |V₊| = |C̄₊| and q + 1 = |V_–| = |C̄_–|. Therefore, either p is even and q is odd or vice versa. Without loss of generality, we assume p as odd and q even.

Suppose that u^′ and u^″ in C̄₊, and v^′ and v^″ in C̄_– have have maximum and minimum modulus values respectively. Then by the same discussion as in (a) with the help of Lemma 4.1, we have

∑uv∈E(C+)XuXv≤∑uv∈E(C¯+)XuXv≤∑uv∈E(B2(p))XuXv, (4.13)

∑uv∈E(C−)XuXv≤∑uv∈E(C¯−)XuXv≤∑uv∈E(B1(q))XuXv, (4.14)

and

∑uv∈E′XuXv≤Xu″Xv″, (4.15)

where the bundle B₂(p) has center u^′ with maximum modulus value and p + 1 = |V₊| ≥ 4 is even. Similarly, the bundle B₁(q) has center v^′, with maximum modulus value and q + 1 = |V₊| ≥ 3 is odd. Now, using (4.13), (4.14) and (4.15) in (4.3), we have

∑uv∈E(C)XuXv≤∑uv∈E(B2(p))XuXv+∑uv∈E(B1(q))XuXv+Xu″Xv″. (4.16)

Since p, q ≥ 2. Therefore, if we take u^″ ∈ B₂(p) and v^″ ∈ B₁(p) such that degree of u^″ in B₂(p) is 1 and degree of v^″ in B₁(q) is 2. Thus, (4.16) becomes

∑uv∈E(C)XuXv≤∑uv∈E(C2(p,q))XuXv. (4.17)

Now by the equations (2.1)-(2.4) and (4.17), λ_min(C₂(p, q)^c) ≤ λ_min(C^c), where n = p + q + 2 > 10 and n ≡ 1(mod 2). Similarly, it also can be prove for all other possibilities.

Now finally we prove that there does not exist any vertex in V₊ such that its value given by X is zero and E^′ has exactly one edge. Firstly, among the vertices of C₁(p, q), we prove that v₂ = u^′ and v₄ = u^″ are unique ones in C₊ and, v₅ = v^″ and v₇ = v^′ are unique ones in C_– with maximum and minimum modulus, respectively. For this, we will show 0 ≤ X₄ < X₃ < X₁ < X₂ and X₇ < X₈ < X₆ < X₅ < 0. By Lemma 4.2, we have X₁, X₂, X₃ non negative and X₄, X₅, X₆, X₇, X₈ negative values in the first eigenvector X of C₁(p, q)^c. By (2.5), λ₁(X₂ – X₁) = –(p – 4)X₁ – X₃ – X₄ < 0, (λ₁ + 1)(X₁ – X₃) = –(X₁ – X₄) < 0, and λ₁(X₃ – X₄) = X₅ < 0 ⇒ X₂ – X₁ > 0, X₁ – X₃ > 0 and X₃ – X₄ > 0. Thus

0≤X4<X3<X1<X2. (4.18)

Similarly, λ₁(X₈ – X₇) = X₅ + X₆ + (q – 4)X₈ < 0, (λ₁ + 1)(X₆ – X₈) = –X₅ + X₈ < 0 and λ₁(X₅ – X₆) = –X₄ < 0 ⇒ X₈ – X₇ > 0, X₅ – X₆ > 0 and X₆ – X₈ > 0. Thus

X7<X8<X6<X5<0. (4.19)

If any one of the vertices v₁, v₂ and v₃ has value zero assigned by X, then by (4.18) X₃ = 0 = X₄. Moreover, by (2.5), we have X₅ = 0 = X₆, which is a contradiction to the construction of V_– and C_–. If the value of the vertex v₄ labeled by X is zero, then by (2.5), λ₁(X₅ – X₆) = 0 ⇒ X₅ = X₆ which is a contradiction to (4.19) (i.e. X₅ is a unique one in C_–). Consequently, X₁, X₂, X₃ and X₄ are non zero positive values of X. Thus, v ∉ V₊ such that X_v = 0. By (4.4), (4.5), (4.8), (4.9) and the above discussion, we have C₊ = C̄₊ = B₁(p), C_– = C̄_– = B₁(q) and E^′ has only one edge u^″v^″ = v₄v₅ in B₁(p, q). Similarly, we can prove for B1′ (p, q) and B₂(p, q). This complete the proof.

Similarly, we can prove the following result:

Lemma 4.4

Suppose that C is a cactus graph of order n = p + q + 2 ≥ 10 such that C^c ∈ Ω1,nc .

If n ≡ 0(mod 2), then either λ_min(C₁(p, q)^c) < λ_min(C^c) or λ_min( C1′ (p, q)^c) < λ_min(C^c).
If n ≡ 1(mod 2), then λ_min(C₂(p, q)^c) < λ_min(C^c).

Theorem 4.5

Suppose that C is a cactus graph of order n such that Cc∈Ωnc=Ω1,nc∪Ω2,nc.

Assume that n ≡ 0(mod 2), p, q ≥ 4 and n = p + q + 2:
1. For p, q ≡ 0(mod 2);
  1. If n ≥ 14 and n ≡ 2(mod 4), then λmin(C1(n−22,n−22)c) ≤ λ_min(C₁(p, q)^c) ≤ λ_min(C^c), where equalities hold if C ≅ C1(n−22,n−22) ≅ λ_min(C₁(p, q)),
  2. If n ≥ 12 and n ≡ 0(mod 4), then λmin(C1(n2,n−42)c) ≤ λ_min(C₁(p, q)^c) ≤ λ_min(C^c), where equalities hold if C ≅ C1(n2,n−42) ≅ λ_min(C₁(p, q)),
2. For p, q ≡ 1(mod 2);
  1. If n ≥ 14 and n ≡ 2(mod 4), then λmin(C1′(n2,n−42)c) ≤ λ_min( C1′ (p, q)^c) ≤ λ_min(C^c), where equalities hold if C ≅ C1′(n2,n−42) ≅ λ_min( C1′ (p, q)),
  2. If n ≥ 12 and n ≡ 0(mod 4), then λmin(C1′(n−22,n−22)c) ≤ λ_min( C1′ (p, q)^c) ≤ λ_min(C^c), where equalities hold if C ≅ C1′(n−22,n−22) ≅ λ_min( C1′ (p, q)).
Assume that n ≡ 1(mod 2), p ≡ 1(mod 2) and q ≡ 0(mod 2):
1. If n ≥ 13 and n ≡ 1(mod 4), then λmin(C2(n+12,n−52)c) ≤ λ_min(C₂(p, q)^c) ≤ λ_min(C^c), where equalities hold if C ≅ C2(n+12,n−52) ≅ λ_min(C₂(p, q)),
2. If n ≥ 15 and n ≡ 3(mod 4), then λ_min (C2(n−12,n−32)c) ≤ λ_min(C₂(p, q)^c) ≤ λ_min(C^c), with equality if C ≅ C2(n−12,n−32) ≅ λ_min(C₂(p, q)).

Proof

The result follows from Lemma 3.3-Lemma 3.5 and Lemma 4.3-Lemma 4.4.

5 Conclusions

Petrović et al. [23] explored a unique cactus as a minimizing graph from the class of cacti such that the order of each cactus is n. But, it is noted that the complement of the proposed minimizing graph is disconnected. In this paper, we characterize the minimizing graphs in a collection of connected graphs such that the complement of each graph of order n is a cactus with the condition that either its each block is only an edge or it has at least one block which is an edge and at least one block which is a cycle. However, the problem is still open to characterize the minimizing graphs in a collection of connected graphs whose complements are in the complete class of cacti (each block of a cactus is only an edge, at least one block is an edge and at least one block is a cycle, or each block is a cycle).

Acknowledgement

The authors are indebted to the anonymous referee for his valuable comments to improve the original version of this paper. The first author is supported by NSFC of China (No.11701530) and the Fundamental Research Funds for the Central Universities (No.2652017146).

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Received: 2018-11-06

Accepted: 2019-08-27

Published Online: 2019-11-28

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

Articles in the same Issue

https://doi.org/10.1515/math-2019-0097

Keywords for this article

adjacency matrix; least eigenvalue; connected graphs; cacti

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