γ-Inverse graph of some mixed graphs

Wafa Boulahmar; Manal Ghanem; Mohammad Abudayah

doi:10.1515/spma-2024-0003

Artikel Open Access

γ-Inverse graph of some mixed graphs

Wafa Boulahmar , Manal Ghanem und Mohammad Abudayah

Veröffentlicht/Copyright: 21. August 2024

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Abstract

Let G be a graph. Then, the inverse graph G − 1 of G is defined to be a graph that has adjacency matrix similar to the inverse of the adjacency matrix of G , where the similarity matrix is ± 1 diagonal matrix. In this article, we introduced a generalization of this definition that serves the mixed graphs where the definition applied for the α -Hermitian adjacency matrices of mixed graphs. Furthermore, for a class of unicyclic graphs, we were able to find an inverse mixed graph for a graph G , where it was proven that G − 1 does not exist.

Keywords: mixed graphs; α-Hermitian adjacency matrix; inverse matrix; bipartite mixed graphs; unicyclic bipartite mixed graphs; perfect matching

MSC 2010: 05C20; 05C50; 15A30

1 Introduction

Let G = ( V ( G ) , E ( G ) ) be an unoriented simple graph, where V ( G ) = { v i } i = 1 n and E ( G ) = { v i v j : v i , v j ∈ V ( G ) } denote the vertex set and the edge set of G , respectively. A w a l k in G is a sequence of vertices, where every two consecutive vertices in the walk form an edge in G . The length of the walk is defined to be number of edges in the walk (including repeated edges). A closed walk is a walk that starts and ends with the same vertex. A cycle is a closed walk of different vertices. A matching of a graph G is a subset M of E ( G ) such that no two edges of M have a common vertex. A matching M is called perfect matching if for every v i ∈ V ( G ) , there is an edge of M incident to v i . A bipartite graph G is a graph where its vertex set V ( G ) can be partitioned into two sets V 1 and V 2 , such that no two vertices of the same set form an edge. Obviously, a graph G is bipartite if and only if G has no odd cycles.

A mixed graph X with underlying graph G = Γ ( X ) is just a graph that can be obtained from G after orienting some or all of its edges. The oriented edges are called arcs, denoted by ordered pairs, and the unoriented edges are called digons, denoted by two consecutive vertices.

Studying a graph structure through different properties of matrices associated with it is a very old and rich area of research. Among the various matrices associated with a graph, the adjacency matrix and Laplacian matrix are probably the most popular and widely investigated ones. The a d j a c e n c y m a t r i x A ( G ) = [ a i j ] of a graph G is the symmetric matrix of size n , where

(1) a i j = 1 , if v i v j ∈ E ( G ) , 0 , otherwise ,

while the Laplacian matrix is defined by L ( G ) = D ( G ) − A ( G ) , where D ( G ) is the diagonal matrix with d i the degree of the vertex corresponding to the i th diagonal entry.

For mixed graphs, recently, there has been a growing study of matrices associated with mixed graphs. The first article that has introduced complex edge weights in oriented graphs was by Bapat et al. [5]. They called it weighted directed graphs; actually, they studied spectral properties of weighted directed graphs with great details. Reff [15] has independently initiated the study of adjacency matrices of graphs with complex edge weights. These concepts are probably the most popular and widely investigated ones. Furthermore, they generalize the adjacency matrix as well as the Laplacian adjacency matrix for mixed graphs (cf. [2,4,6, 9,16] for recent results). Guo and Mohar [8] and [13] have used the concept of the complex unit gain graph to define the α -Hermitian adjacency matrix of mixed graphs as follows:

Given a mixed graph X = ( V ( X ) , E ( X ) ) and a unit complex number α , we define the n × n Hermitian adjacency matrix H α ( X ) = [ h u v ] u , v ∈ V ( X ) by

(2) h u v = 1 , if u v ∈ E ( X ) , α , if ( u , v ) ∈ E ( X ) , α ¯ , if ( v , u ) ∈ E ( X ) , 0 , otherwise .

A perfect matching of a mixed graph is just a perfect matching of its underlying graph. A mixed graph X is called a unicyclic mixed graph if its underlying graph has a unique cycle and it is said to be bipartite if its underlying graph is bipartite. We denote the class of non-bipartite unicyclic mixed graphs with unique perfect matching by U , the class of bipartite unicyclic mixed graphs with unique perfect matching by H , the class of tree mixed graphs with unique perfect matching by T , and the union of the three classes by φ . For X ∈ φ , the unique perfect matching will be denoted by M . An edge of a unicyclic mixed graph X with unique perfect matching is called a peg if it is matching edge and incident to exactly one vertex of the X -cycle C . Finally, for a uncyclic mixed graph X with at least one peg u ′ u , u ′ ∈ V ( C ) , the component of X \ { u ′ } that contains u is called the branch of u and denoted by B u .

The inverse graph was defined in different ways, and the most well-known definition was introduced by Godsil [7] as follows: Let G be a graph and A ( G ) be its adjacency matrix. Then, G is called i n v e r t i b l e if A ( G ) is non-singular, and there is ± 1 diagonal matrix D , such that D A − 1 ( G ) D is an adjacency matrix of another graph G − 1 and the graph G − 1 is called the inverse graph of G . This definition was raised in a motivation by a chemistry problem (for more information, see [14]). In [7], an open problem of characterizing all invertible bipartite graphs was proposed. For example, Akbari and Kirkland [3] characterized all invertible unicyclic bipartite graphs with unique perfect matching. The generalization of the inverse mixed graphs and a related problems were investigated in [10] and [11].

While the inverse mixed graph was lately studied (for example, see [11]), the motivation of this article is to define a γ -inverse mixed graph, X − 1 , of a mixed graph X , where H γ ( X − 1 ) has the same spectrum of the inverse α -hermitian adjacency matrix of X . Actually, this was the main idea behind the inverse graphs that were proposed by Godsil [7]. In fact, we were able to find an inverse mixed graph of a graph G where it was proved in [3] that G does not have an inverse graph.

2 Preliminaries

Given a mixed graph X with its α -Hermitian adjacency matrix H α , suppose that X ′ is a sub-mixed graph of X . Then,

A sub-mixed graph X ′ of X is called elementary sub-mixed graph if the underlying graph of every component of X ′ is isomorphic to either P 2 or C k (for some k ⩾ 3 ).
The rank of X ′ is given by r ( X ′ ) = n − c , where c is the number of its components and n is the number of its vertices.
The co-rank of X ′ is equal to s ( X ′ ) = m − r ( X ′ ) , where m = ∣ E ( X ′ ) ∣ .
A mixed walk W in X is a sequence of vertices v 1 , … , v k of X , where there is an edge or arc between any two consecutive vertices v i , v i + 1 in W . The weight of W is defined as
h α ( W ) = h v 1 v 2 h v 2 v 3 … h v k − 1 v k ∈ { α r } r ∈ Z .
In a graph G with unique perfect matching, a path P is called an mm-alternating path if its edges alternate between matching and non-matching edges whose the first and the last edges is from the matching set. For a mixed graph X with unique perfect matching, a path P in X is called mm-alternating if its underlying path is mm-alternating.

A graph is defined to be singular and nonsingular based on the singularity of its adjacency matrix. In order to define singular (nonsingular ) mixed graph, consider the cycle graph C 8 and the mixed cycle C 8 , 1 obtained from C 8 by orienting one edge. One can easily check that the adjacency matrix of C 8 is singular, while the α -hermitian adjacency matrix of C 8 , 1 is nonsingular in general, which means that the singularity of the hermitian adjacency matrix of a mixed graph depends on the value of α .

Definition 1

A mixed graph X is called α -singular (resp. α -non-singular) if its α -Hermitian adjacency matrix H α ( X ) is singular (resp. non-singular).

Definition 2

Let X be an α -non-singular mixed graph with its α -Hermitian adjacency matrix H α ( X ) . If there exists a diagonal matrix Δ with diagonal entries ± α k such that k is an integer number and Δ H α − 1 ( X ) Δ * is a γ -Hermitian adjacency matrix of a mixed graph X − 1 , the mixed graph X − 1 is called the γ -inverse mixed graph of X .

Note that the γ -inverse mixed graphs can be considered as a natural generalization of the classical definition of the inverse graphs. Furthermore, we will see later on in this article that some mixed graphs have inverse graphs, which means, in the contrast of the classical definition, that the non-existence of ± 1 diagonal matrix Δ does not mean that the graph have no inverse, but it may have inverse mixed graph.

Recall that a bijective function σ of a set of n -elements is called a permutation. The set of all permutations of a set V with n -elements together with functions composition forms a group, denoted by S n . The sign of a permutation σ , sgn ( σ ) , is defined as ( − 1 ) k , where k is the number of transpositions when σ is decomposed as a product of transpositions. The following theorem is well known as a classical result in linear algebra.

Theorem 1

If A = [ a i j ] is an n × n matrix, then

det ( A ) = ∑ σ ∈ S n sgn ( σ ) a 1 , σ ( 1 ) a 2 , σ ( 2 ) … a n , σ ( n ) .

Using Theorem 1, authors in [1] proved the following theorem.

Theorem 2

Let X be a mixed graph and H α ( X ) be its α -Hermitian adjacency matrix. Then,

det ( H α ) = ∑ X ′ ( − 1 ) r ( X ′ ) 2 s ( X ′ ) Re ( ∏ C h α ( C → ) ) ,

where the sum ranges over all spanning elementary sub-mixed graphs X ′ of X, the product ranges over all mixed cycles C in X ′ and C → is any mixed closed walk traversing C .

3 Inverse of the α -Hermitian adjacency matrix of φ

In order to study the inverse of the α -Hermitian adjacency matrix of a mixed graph X ∈ φ , we need to study whether the α -Hermitian adjacency matrix of such mixed graph is singular or not. As we mentioned before, the singularity of α -Hermitian adjacency matrix of a mixed graph depends (sometimes) on the value of α . But using Theorem 2, one can get that the determinant of α -Hermitian adjacency matrix of a mixed graph X can only depend on α when its underlying graph has a spanning elementary mixed graph X ′ , with one of its components is a cycle. Otherwise, the determinant of the α -Hermitian adjacency matrix of a mixed graph X will be equal to the determinant of the adjacency matrix of Γ ( X ) . Therefore, for a mixed graph X with unique spanning elementary sub-mixed graph that consists of arcs and digons, we have

det ( H α ( X ) ) = det A ( Γ ( X ) ) = ( − 1 ) n 2 , n = ∣ V ( X ) ∣ .

Now, for a unicyclic mixed graph X with pegs and with unique perfect matching, we have the following cases:

X is a nonbipartite mixed graph. If X has a spanning elementary sub-mixed graph X ′ consisting of the mixed cycle C , and independent arcs and digons, then it is observed that C is an odd cycle and X ′ has an odd number of vertices. This contradicts the fact that X has a perfect matching.
X is a bipartite mixed graph. If X has a spanning elementary sub-mixed graph X ′ consisting of the mixed cycle C , and independent arcs and digons, then the trees incident to the cycle through the pegs will have odd number of vertices, which is a contradiction.

The following theorem is an immediate consequence of the aforementioned discussion.

Theorem 3

Let X ∈ φ with ∣ V ( X ) ∣ = n . Then, for any unit complex number α , we have

det ( H α ( X ) ) = ( − 1 ) n 2 .

Now, we are ready to study the entries of the inverse of the α -Hermitian adjacency matrix of a mixed graph X ∈ φ . In fact, we will give a complete description of the entries in terms of the mixed paths between vertices.

Theorem 4

Let X ∈ φ . Then,

[ H α − 1 ] i j = ∑ P i → j ( − 1 ) n 2 + ∣ E ( P i → j ) ∣ h α ( P i → j ) det ( X ¯ ) , if i ≠ j , ( − 1 ) n 2 det ( ( H α ) ( i , i ) ) , if i = j ,

where P i → j is a path from i to j and X ¯ = X \ P i → j .

Proof

Let X ∈ φ and H α ( X ) be its α -Hermitian adjacency matrix. Then, using Cramer’s rule,

[ H α − 1 ] i j = c i j det ( H α ) ,

where c i j = ( − 1 ) i + j det ( ( H α ) ( j , i ) ¯ ) and ( H α ) ( j , i ) is the matrix obtained from H α ( X ) after removing the j th row and the i th column. Let M j i be the matrix obtained from H α ( X ) by replacing the ( j i ) -entry by 1 and all other entries in the j th row and i th column by 0. Then, c i j = det ( M j i ) ¯ .

If i = j , then det ( M i i ) = det ( ( H α ) ( i , i ) ) . So using Theorem 3,
[ H α − 1 ] i i = c i i det ( H α ) = det ( ( H α ) ( i , i ) ) det ( H α ) = ( − 1 ) n 2 det ( ( H α ) ( i , i ) ) .
If i ≠ j , using Theorem 1,
det ( M j i ) = ∑ π ∈ S n sgn ( π ) ∏ k = 1 n m k , π ( k ) .
Now, by construction of M j i , only permutation such that π ( j ) = i contributes to non-zero term in the expansion of det ( M j i ) . Let S j → i be the family of all permutations on V ( X ) such that π ( j ) = i . Denote the cycle of π permuting j to i by σ π and all other cycles by σ π c , further let P π (resp., P π c ) be the set of vertices in σ π (resp., σ π c ), then:
det ( M j i ) = ∑ π ∈ S j → i sgn ( π ) ∏ k ∈ V ( X ) \ { j } m k , π ( k ) = ∑ π ∈ S j → i ( sgn ( σ π ) ∏ k ∈ P π m k , σ π ( k ) ) ( sgn ( σ π c ) ∏ k ∈ P π c m k , σ π c ( k ) ) .
Let P j → i be a path from j to i following the permutation order in σ π . Then,
sgn ( σ π ) ∏ k ∈ P π m k , σ π ( k ) = ( − 1 ) ∣ E ( P j → i ) ∣ h α ( P j → i ) .
Now, let Φ be the set of all cycles σ ∈ S n that permute j to i , and let Φ σ be the collection of all permutations π ∈ S n such that σ is one of its cycles. Then,
∑ π ∈ S j → i ( sgn ( σ π ) ∏ k ∈ P π m k , σ π ( k ) ) ( sgn ( σ π c ) ∏ k ∈ P π c m k , σ π c ( k ) ) = ∑ σ ∈ Φ ∑ π ∈ ϕ σ ( sgn ( σ π ) ∏ k ∈ P π m k , σ π ( k ) ) ( sgn ( σ π c ) ∏ k ∈ P π c m k , σ π c ( k ) ) = ∑ σ ∈ Φ ( − 1 ) ∣ E ( P j → i ) ∣ h α ( P j → i ) ∑ π ∈ ϕ σ ( sgn ( σ π c ) ∏ k ∈ P π c m k , σ π c ( k ) ) = ∑ σ ∈ Φ ( − 1 ) ∣ E ( P j → i ) ∣ h α ( P j → i ) det ( H α ( X ¯ ) ) ,
where X ¯ is the graph obtained from X after removing all vertices of P j → i . Finally, being c i j = det ( M j i ) ¯ , we obtain the result.□

First, we will deal with the diagonal entries of the inverse of α -Hermitian adjacency matrix of a mixed graph X ∈ φ , which is mentioned in Theorem 4. Obviously, if X is bipartite, then det ( ( H α ) ( i , i ) ) should be zero, for every vertex i of X . Thus, we need only to deal with the case of non-bipatite unicyclic mixed graphs.

Lemma 1

For a mixed graph X ∈ U , there is a vertex i of X such that [ H α − 1 ] i i ≠ 0 if and only if X has only one peg.

Proof

Suppose that [ H α − 1 ] i i ≠ 0 . Then, using Theorem 4, det ( ( H α ) ( i , i ) ) ≠ 0 . Thus, X \ { i } has elementary sub-mixed graph ( X \ { i } ) ′ . Furthermore, since the number of vertices of X is even, ( X \ { i } ) ′ is unique and consists of the unique cycle of X , C , and independent arcs and digons. Now, suppose that X has more than one peg, then the independent arcs and digons of ( X \ { i } ) ′ will contain none of these pegs. Let u u ′ be one of the pegs of X \ { i } , u ′ ∈ V ( C ) . This means that the branch B u has a perfect matching and the tree B u ∪ { u ′ } also has a perfect matching, which is a contradiction.

Conversely, suppose that X has only one peg, say u u ′ , where u ′ ∈ V ( C ) . Then, obviously, X \ { u } will contain two components B u \ { u } and X \ B u . Now, X \ B u has elementary sub-mixed graph consisting of the cycle and the perfect matching of trees that are incident to the cycle. Furthermore, B u \ { u } has a perfect matching. Therefore, X \ { u } has an elementary sub-mixed graph. Since this elementary should contain one cycle and edges, this elementary sub-mixed graph is unique, and thus, [ H α − 1 ] i i ≠ 0 .□

Now, note that the non-diagonal entry of the inverse of α -Hermitian adjacency matrix of a mixed graph, say [ H α − 1 ] i j , depends on the determinant of the α -Hermitian adjacency matrix of the mixed graph X \ P i → j . To be more specific, this determinant determines whether the term corresponds to the path P i → j is zero or not. Note also that since X ∈ φ , X \ P i → j is a unicyclic mixed graph or a forest. In the following, we will study when X \ P i → j has a unique perfect matching and when does not.

Theorem 5

Let X ∈ U with more than one peg. Then, P u → v is an mm-alternating path from u to v if and only if X \ P u → v has a unique elementary sub-mixed graph.

Proof

Let X ∈ U with more than one peg. Suppose that P u → v is an mm-alternating path from u to v , then all P u → v vertices will be matched by matching edges M ∩ E ( P u → v ) = M 1 , which means no matching edges from M \ E ( P u → v ) will incident to P u → v vertices. Therefore, M \ M 1 will form a perfect matching for X \ P u → v .

Now, assume that X \ P u → v has more than one elementary sub-mixed graph M \ P u → v say ( X \ P u → v ) ′ . If C is one of ( X \ P u → v ) ′ components, then X \ P u → v has an odd number of vertices, which contradicts the fact that both X and P u → v have even number of vertices. Therefore, ( X \ P u → v ) ′ form a perfect matching of X \ P u → v . Thus, ( X \ P u → v ) ′ ∪ M 1 ≠ M form a perfect matching of X , which is a contradiction.

Conversely, suppose that X \ P u → v has unique elementary sub-mixed graph ( X \ P u → v ) ′ . Then, we claim that ( X \ P u → v ) ′ forms a perfect matching of X \ P u → v . Indeed, the claim is obvious when u and v are not vertices of the same branches. Suppose that u and v are two vertices of same branch B w , C is the cycle component of X ′ , and B s is the another branch of X . Then, since C is a component of ( X \ P u → v ) ′ , we have B s that has a perfect matching, but since s s ′ is a peg of X , we have B s ∪ { s ′ } that has a perfect matching, which is a contradiction.

Now, suppose that P u → v is not mm-alternating path and X \ P u → v is a sub-mixed graph with unique perfect matching. Let r be a vertex of P u → v that is incident to a matching edge r r ′ from outside of M ∩ P u → v , where r ∈ P u → v . Let Y be the sub-mixed component of X \ P u → v that contains r ′ . Then, if Y is a unicyclic graph, then Y has a perfect matching and Y ∪ { r } has a perfect matching, which is a contradiction.□

Theorem 6

Let X ∈ H be a mixed graph with more than two pegs and u and v be two vertices in X. Then, for a path P u → v , X \ P u → v has a unique elementary spanning sub-mixed graph if and only if P u → v is an mm-alternating path.

Proof

Suppose that X \ P u → v has a unique elementary sub-mixed graph. Observing that X and X \ P u → v have even number of vertices and P u → v has even number of vertices. Now, suppose that P u → v is not an mm-alternating path, then there are at least two vertices r and s of P u → v such that r and s are matched by two matching edges r r ′ , s s ′ ∈ M \ E ( P u → v ) . Obviously, r ′ and s ′ belong to the component in X \ P u → v that includes the X -cycle C . Therefore, X contains more than one cycle, which is a contradiction.

Conversely, let P u → v be an mm-alternating path from u to v . Suppose that X \ P u → v has more than one elementary sub-mixed graph ( X \ P u → v ) ′ and ( X \ P u → v ) ′ ′ . Then, since X has more than two pegs, the X cycle will not be a component of ( X \ P u → v ) ′ and ( X \ P u → v ) ′ ′ , and thus, they will form two different perfect matchings of X \ P u → v . Finally, since P u → v is an mm-alternating path, ( P u → v ∩ M ) ∪ ( X \ P u → v ) ′ and ( P u → v ∩ M ) ∪ ( X \ P u → v ) ′ ′ will form two different perfect matchings of X , which is a contradiction.□

Corollary 1

Let X ∈ φ with more than two pegs. Then,

[ H α − 1 ] i j = ∑ P i → j ∈ F ( − 1 ) ∣ E ( P i → j ) ∣ − 1 2 h α ( P i → j ) , if i ≠ j , 0 , otherwise ,

where F = { P i → j : P i → j is a n mm - alternating p a t h f r o m i to j } .

Proof

Using Theorem 4 together with Lemma 1, Theorems 5, and 6 one can obtain:

[ H α − 1 ] i j = ∑ P i → j ∈ F i → j ( − 1 ) n 2 + ∣ E ( P i → j ) ∣ h α ( P i → j ) det ( X ¯ ) , if i ≠ j , 0 , otherwise ,

where F i → j is the set of all mm-alternating paths from i to j and X ¯ is the graph obtained from X after removing all vertices of P i → j . Finally, using the fact that X ¯ ∈ φ together with Theorem 3, we obtain

det ( X ¯ ) = ( − 1 ) n − ∣ E ( P i → j ) − 1 2 ,

which ends the proof.□

The formula in Corrollary 1 can be simplified using the following lemmas.

Lemma 2

Let X ∈ U . Then, for any two vertices u and v of X, there is at most one mm-alternating path from u to v.

Proof

Suppose that there are two mm-alternating paths from u to v say P u → v and P u → v ′ . Since each of u and v is incident to exactly one matching edge, u and v should not be the vertices of the X -cycle C , which means V ( C ) ⊂ ( V ( P u → v ) ∪ V ( P u → v ′ ) ) . Observing that X has unique perfect matching, C has an odd number of pegs, and thus, there is a peg that is incident to a vertex of P u → v or P u → v ′ . This contradicts being both P u → v and P u → v ′ mm-alternating paths.□

The following lemma can be found in [1].

Lemma 3

Let X ∈ H . If X has more than two pegs, then for any two vertices u, v of X, there is at most one mm-alternating path from u to v.

Corollary 2

Let X ∈ φ with more than two pegs. Then,

[ H α − 1 ] i j = ( − 1 ) ∣ E ( P i → j ) ∣ − 1 2 h α ( P i → j ) , if P i → j is a n mm - alternating p a t h f r o m i to j , 0 , otherwise .

It should be mentioned here that Corollary 2 is not true when a uni-cyclic mixed graph X has only one peg. For example consider the uni-cyclic mixed graph X in Figure 1. Obviously there is no mm-alternating path between the vertex 12 and 14, however its easy to check that the entry [ H − 1 ] { 12 14 } of [ H − 1 ] is not zero. The Following theorem take care of this case.

$Figure 1 There is no mm-alternating path between the vertices 12 and 14; however, P 12 → 14 {P}_{12\to 14} contributes to non-zero value in H α − 1 {H}_{\alpha }^{-1} .$

Figure 1

There is no mm-alternating path between the vertices 12 and 14; however, P 12 → 14 contributes to non-zero value in H α − 1 .

Theorem 7

Let X ∈ U with one peg, then [ H α − 1 ] i j ≠ 0 if and only if there is an mm-alternating path from i to j or there is an mm-alternating walk through the cycle and the one peg from i to j .

Proof

Suppose that [ H α − 1 ] i j ≠ 0 , and then using Theorem 4, there is P i → j such that X \ P i → j has elementary sub-mixed graph.

Case 1: P i → j is of even number of vertices.
Suppose that P i → j is not an mm-alternating path. Then, there are two vertices, say v 1 , v 2 ∈ P i → j , such that v 1 and v 2 are incident to two matching edges v 1 v 1 ′ , v 2 v 2 ′ with the property v 1 v 1 ′ , v 2 v 2 ′ ∉ E ( P i → j ) .
Let G 1 (resp. G 2 ) be the component of X \ P i → j that contains v 1 ′ (resp. v 2 ′ ). Since X is a unicyclic mixed graph, at least one of the induced sub-graphs of X over V ( G 1 ) ∪ { v 1 } = D 1 or V ( G 2 ) ∪ { v 2 } = D 2 will be a tree and D 1 and D 2 should be disjoint; otherwise, the mixed graph, X will have two cycles. Assume that G 1 is a tree with perfect matching, as X \ P i → j has elementary sub-mixed graph for each component, then the number of vertices of G 1 is even. But V ( G 1 ) ∪ { v 1 } = D 1 has a spanning elementary sub-mixed graph, which is perfect matching with even vertices, which is a contradiction.
Case 2: P i → j is of odd number of vertices.
Let u ′ u 1 be the cycle peg, where u ′ ∈ V ( C ) ; suppose that l u ′ i : u ′ , u 1 , u 2 , … , ( u k = i ) (resp. J u ′ j ) be the path from u ′ to i (resp. from u ′ to j ). Then, since X is of even vertices, any spanning elementary mixed sub-graph of X \ P i → j should contain the cycle as one of its components. So, there is v ∈ V ( X ) such that u 1 v is an edge in the spanning elementary sub-mixed graph of X \ P i → j . If v ≠ u 2 , then the tree component of X \ P i → j that contains v , say T v , and T v ∪ { u 1 } have perfect matching, which is contradiction. Therefore, v = u 2 . Note here that u ′ u 1 is a matching edge, and so, u 1 u 2 is not a matching edge of X . Therefore, there is v 1 ∈ V ( X ) such that u 2 v 1 is a matching edge. If v 1 ≠ u 3 , then since X \ P i → j has a spanning elementary sub-mixed graph with a cycle component, u 1 u 2 is an edge in this elementary sub-mixed graph and u 2 v 1 will not be a part of the elementary sub-mixed graph. Therefore, the connected tree T v 1 of X \ ( P i → j ∪ ( u 1 u 2 ) ) that contains v 1 has a perfect matching, but T v 1 ∪ { u 2 } has a perfect matching, which is a contradiction. So v 1 = u 3 continue in this way until u k = i . The same thing can be done for J u ′ j . Thus, the walk l i → u ′ ∪ C ∪ J u ′ → j forms an mm-alternating walk from i to j that passes through the cycle C .

Conversely, suppose that there is an mm-alternating path from i to j ; then using Theorem 4 and Lemma 2, the result holds.

Now, suppose that there is an mm-alternating walk W i → j that passes through the cycle and the peg. Then,

Claim: X \ P i → j , P i → j is a path from i to j and has a unique spanning elementary sub-mixed graph.

Indeed, let u u ′ be the X peg, where u ′ ∈ V ( C ) , and v be the vertex of P i → j with minimum distance with u ′ .

Then, since W i → j is mm-alternating, W i → j = P i → v ∪ P v → u ′ ∪ C ∪ P u ′ → v ∪ P v → j , and thus, it contains even number of vertices and P i → j = P i → v ∪ P v → j . But the number of vertices of P v → u ′ ∪ P u ′ → v is even so the number of vertices of P i → j should be odd. Thus, there is at least a vertex t of P i → j that is matched by a matching edge that lies outside the path P i → j .

Sub-claim: t is unique and t = v .

Suppose that t ≠ v and t t ′ is the matching edge with t t ′ ∉ E ( P i → j ) . Then, t t ′ is not an edge of X \ P i → j , so the tree component T t ′ of X \ P i → j that contains t ′ has a perfect matching, but T t ′ ∪ { t } has a perfect matching, which is a contradiction.

Now, consider M 1 = E ( P u ′ → v ) \ M and M 2 = M \ ( E ( P i → j ) ∪ E ( C ) ∪ E ( P u ′ → v ) ) . Then, M 1 ∪ M 2 ∪ C will form a spanning elementary sub-mixed graph of X \ P i → j .

Assume that there exists another elementary sub-mixed graph of X \ P i → j , then C must be a component of it. Therefore, X \ ( P i → j ∪ C ) will have two different perfect matchings, which is a contradiction.□

4 γ -Inverse mixed graph of X ∈ φ

In Section 3, we calculated the entries of the inverse matrix of the α -Hermitian adjacency matrix of a mixed graph X in terms of the paths between vertices. Recall that a mixed graph is called α -invertible if there is a diagonal matrix Δ = diag [ { ± α r i } i = 1 n ] such that Δ H α − 1 ( X ) Δ * is a γ -Hermitian adjacency matrix of a mixed graph X − 1 . In this section, we will investigate when such Δ exists and when does not.

Bapat and Kalita [5] intiated a new technique to define a diagonal matrix that transforms a mixed graph G to another H . In this case, G and H are called D -similar. We will use the same technique in this section with more details and restrictions that help us to reduce the number of oriented edges in a unicyclic mixed graph. Let X be a connected mixed graph, u be a vertex of X and W : ( u = u 1 ) and u 2 , … , u k be a walk in X . Recursively, we define a function that assigns a value f W , u α ( j ) to the j th vertex along W as

f W , u α ( 1 ) = 1

and

f W , u α ( j + 1 ) = f W , u α ( j ) , if u j u j + 1 is a digon in X , α f W , u α ( j ) , if u j u j + 1 is an arc in X , α ¯ f W , u α ( j ) , if u j + 1 u j is an arc in X ,

where j = 1 , … , k − 1 , we will write f W , u α ( * ) for the final value f W , u α ( k ) . Let F u = { f W , u α ( * ) : W is a walk in X with starting vertex u } .

Now, we define the α -store of u as

S α ( u ) = { f W , u α ( * ) : W is a closed walk in X starting at u } .

For a vertex v , we define the α -store of the vertex v with respect to the vertex u as

T u α ( v ) = { f W , u α ( * ) : W is a walk from u to v } ,

and the store size as

s α ( u ) = ∣ S α ( u ) ∣ .

It is obvious that 1 ∈ S α ( u ) . If X contains a cycle of weight other than 1, then along any walk W that contains C the store of u will contain values other than 1. If α is a primitive n th root of unity, and the weight of the cycle C is α k , where g c d ( n , k ) = 1 , then the store of the vertex u will contain all powers of α .

The next result is mentioned for the importance of its conditions. The proof can be found in [12].

Lemma 4

Let X be a connected mixed graph. Then, the following statements are equivalent:

s u α ( v ) = 1 , for at least one vertex v ∈ V ( X ) .
s u α ( v ) = 1 , for every vertex v ∈ V ( X ) .
h α ( C ) = 1 , for every cycle C in X .
f W 1 , u α ( * ) = f W 2 , u α ( * ) , for every pair W 1 and W 2 of walks in X having the same end vertex.

Definition 3

A connected mixed graph X is called an α -stable graph if h α ( C ) = 1 for every cycle C in X .

The aforementioned definition is a special case of stable weighted directed graph, which was introduced in [5]. Furthermore, they developed the “D-similarity” concept, which plays the rule of converting the weighted directed graph to its underlying graph through similarity of their adjacency matrices. In the following, we will develop a similar concept to serve the non- α -stable mixed graphs but in different technique.

Let X ∈ φ be a unicyclic mixed graph with the vertex set { v i } i = 1 n . Then, for every edge e of the X -cycle C , X \ e will form a tree. Using the fact that for every tree T , T is α -stable, we can define the diagonal matrix

Δ α , u e = diag { f P u → v 1 , u α ( v 1 ) , f P u → v 2 , u α ( v 2 ) , … , f P u → v n , u α ( v n ) } ,

for some u ∈ V ( X ) . The following theorem illustrates the importance of the diagonal matrix Δ α , u e .

Theorem 8

Let X ∈ φ be a unicyclic mixed graph and e be an edge of the X -cycle C. Then, for any vertex u of X, we have

Δ α , u e H α ( X ) ( Δ α , u e ) * = H γ ( Y e ) ,

where Y e is the mixed graph obtained from X after converting all directed edges of X to digons except, possibly e, and γ = h α ( C ) .

Proof

Assume that X ∈ φ be a unicyclic mixed graph and e be an edge of the X -cycle C . Then, since X \ e is a tree mixed graph, we have the following:

For any edge i j ∈ X \ e ,

(3) [ Δ α , u e H α ( X ) ( Δ α , u e ) * ] i j = f P u → i , u α ( i ) . [ H α ( X ) ] i j . f P u → j , u α ( j ) ¯

(4) = f P u → i , u , u α ( i ) . h i j . f P u → j , u α ( j ) ¯

(5) = f P u → i , u , u α ( i ) f P u → j , u α ( j ) . h i j

(6) = 1 .

Now, suppose that e = r s . Then, observing that X \ e is a tree, the store value of any vertex of X will be fixed. Now, let P u → r and P u → s ) be two paths from u to r and from u to s , respectively. Then, obviously, h α ( C ) = h α ( P u → r ) h α ( r s ) h α ( P u → s ) ¯ . Therefore,

(7) h α ( C ) = h α ( P u → r ) h α ( r s ) h α ( P u → s ) ¯

(8) = f P u → r , u α ( r ) h α ( r s ) f P u → s , u α ( s ) ¯

(9)□ = [ Δ α , u e H α ( Δ α , u e ) * ] r s .

Now, using Theorem 8,

Δ α , u e H α − 1 ( X ) ( Δ α , u e ) * = H γ − 1 ( Y e ) .

Therefore, for a mixed graph X ∈ φ with more than two pegs, all edges of the mixed graph Y e are digons except, probably e , and using Corollary 2, one can easily deduce that the entries of H γ − 1 ( Y e ) are ± γ , ± γ ¯ , or ± 1 . The following theorem will care about the sign of the entries of H γ − 1 ( Y e ) .

Theorem 9

For a mixed graph X ∈ φ , if X has more than two pegs, then X is α -invertible.

Proof

Let e be an edge of the X -cycle C and u be a vertex of X . Then,

Δ α , u e H α − 1 ( X ) ( Δ α , u e ) * = H γ − 1 ( Y e ) ,

where Y e is the mixed graph obtained from X after converting all directed edges of X to digons except, possibly e , and γ = h α ( C ) . Now, let Z e be the mixed graph obtained from Γ ( Y e ) after orienting all non-matching edges. Then, obviously,

H γ = [ h i j ] = Δ ( − 1 ) , u e H γ ( Y e ) Δ ( − 1 ) , u e

will be the adjacency matrix of the weighted mixed graph obtained from Y e by multiplying the weight of each nonmatching edge e ′ ( e ′ ≠ e ) by − 1 . Now, applying Corollary 2, we obtain

[ Δ ( − 1 ) , u e H γ ( Y e ) Δ ( − 1 ) , u e ] i j − 1 = h γ ( P i → j ) , if P i → j is an mm-alternating path from i to j in Y e , 0 , otherwise .

Thus, all entries of H γ − 1 will be from the set { 0 , 1 , γ , γ ¯ } or all from the set { 0 , 1 , − γ , − γ ¯ } , which ends the proof (Figure 2).

$Figure 2 The transformation of the mixed graph X X through the operators Δ α , u e {\Delta }_{\alpha ,u}^{e} and Δ ( − 1 ) , u e Δ α , u e {\Delta }_{\left(-1),u}^{e}{\Delta }_{\alpha ,u}^{e} .$

Figure 2

The transformation of the mixed graph X through the operators Δ α , u e and Δ ( − 1 ) , u e Δ α , u e .

Finally, it should be mentioned here that one can construct the diagonal matrix Δ ( − 1 ) , u e Δ α , u e immediately using the function g W , u defined below. Let X ∈ φ and u ∈ V ( X ) . Then, for a walk W : u , u 2 , … , u k in X , define a function that assigns the value g W , u ( j ) for the j th vertex as follows:

g W , u α ( 1 ) = 1

and

(10) g W , u α ( j + 1 ) = − f W , u α ( j + 1 ) , if u j u j + 1 is a non matching arc or digon in X , f W , u α ( j + 1 ) , if u j u j + 1 is a matching arc or digon in X .

Example 1

Consider the mixed graph X and the sub-mixed graph X ′ represented by the blue arcs in Figure 3(a). According to the function g W , 1 α the diagonal matrix Δ g is as follows:

Δ g = 1 0 0 0 0 0 0 α 0 0 0 0 0 0 − α 0 0 0 0 0 0 − α 2 0 0 0 0 0 0 1 0 0 0 0 0 0 α .

The weighted mixed graph corresponds to the Hermitian adjacency matrix H γ , which is shown in Figure 3(b), and finally, the mixed graph in Figure 3(c) is an inverse mixed graph of X .

$Figure 3 (a) Mixed graph X with the values of g W , 1 α {g}_{W,1}^{\alpha } , (b) the weighted mixed graph that corresponds to the Hermitian adjacency matrix H γ {{\mathscr{H}}}_{\gamma } , and (c) an α 2 {\alpha }^{2} -inverse mixed graph of X X .$

Figure 3

(a) Mixed graph X with the values of g W , 1 α , (b) the weighted mixed graph that corresponds to the Hermitian adjacency matrix H γ , and (c) an α 2 -inverse mixed graph of X .

Theorem 10

[3] Let G be a bipartite unicyclic graph having a unique perfect matching. Let the cycle be of length 2 m , and suppose that there are 2 k pegs. Let A be the adjacency matrix of G. Then,

A − 1 is signable to a ( 0 , 1 ) -matrix (i.e., there is a ( ± ) 1 -diagonal matrix D such that D A − 1 D is a ( 0 , 1 ) -matrix) if and only if one of the following holds:
1. k ≥ 2 and m − k is even,
2. k = 1 , m is even and the vertices on the cycle incident with pegs are adjacent.
A − 1 is signable to a non-negative matrix if and only if either of conditions ( a ) and ( b ) hold, or ( c ) k = 1 and m is odd.

Note here based on Theorem 10, the underlying graph of the mixed graph Y shown in Figure 4(c) has no inverse graph. However, the following example illustrates that it has an inverse mixed graph.

$Figure 4 (a) A non- α \alpha -stable mixed graph X X , (b) a non- { − 1 } \left\{-1\right\} -stable mixed graph X ′ X^{\prime} , and (c) the mixed graph Y Y that corresponds to the inverse of the { − 1 } \left\{-1\right\} -Hermitian adjacency matrix of X ′ X^{\prime} .$

Figure 4

(a) A non- α -stable mixed graph X , (b) a non- { − 1 } -stable mixed graph X ′ , and (c) the mixed graph Y that corresponds to the inverse of the { − 1 } -Hermitian adjacency matrix of X ′ .

Example 2

Consider the non- α -stable mixed graph X shown in Figure 4(a) together with α = e π 4 i . Obviously, the mixed graph Y = X \ e 67 is the balanced graph. Therefore, the matrix

Δ α , 0 H α ( X ) Δ α , 0 *

will form the { − 1 } -Hermitian adjacency matrix of the mixed graph X ′ shown in Figure 4(b), where

Δ α , 0 = 1 0 0 0 0 0 0 0 0 0 0 α 3 0 0 0 0 0 0 0 0 0 0 α 2 0 0 0 0 0 0 0 0 0 0 α 2 0 0 0 0 0 0 0 0 0 0 α 2 0 0 0 0 0 0 0 0 0 0 α 0 0 0 0 0 0 0 0 0 0 α 3 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 α 0 0 0 0 0 0 0 0 0 0 α 2 .

The inverse of Δ α , 0 H α Δ α , 0 * will form the adjacency matrix of the weighted mixed graph as shown in Figure 4(c). One can easily check that the { − 1 } -Hermitian adjacency matrix is similar to the adjacency matrix of its underlying graph through the diagonal matrix:

Δ − 1 , 0 = 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 − 1 0 0 0 0 0 0 0 0 0 0 − 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 − 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 − 1 0 0 0 0 0 0 0 0 0 0 1 .

Funding information: The authors state no funding involved.
Author contribution: All authors have taken responsibility for the complete content of this manuscript and have given their consent for its submission to the journal. They reviewed all results and approved the final version of the manuscript. Prof. Mohammad proposed the paper’s concept, while Prof. Mohammad and Prof. Manal contributed to the literature review and demonstrated theorems related to cospectrality. Wafa applied the cospectrality concept to extend and generalize the inverse graph theory.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no data sets were generated or analyzed during this study.

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Received: 2023-10-18

Revised: 2024-03-15

Accepted: 2024-03-15

Published Online: 2024-08-21

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https://doi.org/10.1515/spma-2024-0003

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mixed graphs; α-Hermitian adjacency matrix; inverse matrix; bipartite mixed graphs; unicyclic bipartite mixed graphs; perfect matching

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