In [Steve Butler. A note about cospectral graphs for the adjacency and normalized Laplacian matrices. Linear Multilinear Algebra, 58(3-4):387-390, 2010.], Butler constructed a family of bipartite graphs, which are cospectral for both the adjacency and the normalized Laplacian matrices. In this article, we extend this construction for generating larger classes of bipartite graphs, which are cospectral for both the adjacency and the normalized Laplacian matrices. Also, we provide a couple of constructions of non-bipartite graphs, which are cospectral for the adjacency matrices but not necessarily for the normalized Laplacian matrices.
Let $Γ=(G,σ)$ be a signed graph, where $σ$ is the sign function on the edges of $G$. In this paper, we use the operation of partial transpose to obtain non-isomorphic Laplacian cospectral signed graphs. We will introduce two new operations on signed graphs. These operations will establish a relationship between the adjacency spectrum of one signed graph with the Laplacian spectrum of another signed graph. As an application, these new operations will be utilized to construct several pairs of cospectral non-isomorphic signed graphs. Finally, we construct integral signed graphs.
The spectrum of the normalized Laplacian matrix cannot determine the number of edges in a graph, however finding constructions of cospectral graphs with differing number of edges has been elusive. In this paper we use basic properties of twins and scaling to show how to construct such graphs. We also give examples of families of graphs which are cospectral with a subgraph for the normalized Laplacian matrix.
We give a construction of a family of (weighted) graphs that are pairwise cospectral with respect to the normalized Laplacian matrix, or equivalently probability transition matrix. This construction can be used to form pairs of cospectral graphs with differing number of edges, including situations where one graph is a subgraph of the other. The method used to demonstrate cospectrality is by showing the characteristic polynomials are equal.
Let $n$ be any positive integer and let $F_n$ be the friendship (or Dutch windmill) graph with $2n+1$ vertices and $3n$ edges. Here we study graphs with the same adjacency spectrum as the $F_n$. Two graphs are called cospectral if the eigenvalues multiset of their adjacency matrices are the same. Let $G$ be a graph cospectral with $F_n$. Here we prove that if $G$ has no cycle of length 4 or 5, then $G\cong F_n$. Moreover if $G$ is connected and planar then $G\cong F_n$. All but one of connected components of $G$ are isomorphic to $K_2$. The complement $\bar{F_n}$ of the friendship graph is det…
We give a method to construct cospectral graphs for the normalized Laplacian by a local modification in some graphs with special structure. Namely, under some simple assumptions, we can replace a small bipartite graph with a cospectral mate without changing the spectrum of the entire graph. We also consider a related result for swapping out biregular bipartite graphs for the matrix $A+tD$. We produce (exponentially) large families of non-bipartite, non-regular graphs which are mutually cospectral, and also give an example of a graph which is cospectral with its complement but is not self-compl…
Graphs can be associated with a matrix according to some rule and we can find the spectrum of a graph with respect to that matrix. Two graphs are cospectral if they have the same spectrum. Constructions of cospectral graphs help us establish patterns about structural information not preserved by the spectrum. We generalize a construction for cospectral graphs previously given for the distance Laplacian matrix to a larger family of graphs. In addition, we show that with appropriate assumptions this generalized construction extends to the adjacency matrix, combinatorial Laplacian matrix, signles…
The distance matrix $\mathcal{D}(G)$ of a graph $G$ is the matrix containing the pairwise distances between vertices, and the distance Laplacian matrix is $\mathcal{D}^L(G)=T(G)-\mathcal{D}(G)$, where $T(G)$ is the diagonal matrix of row sums of $\mathcal{D}(G)$. We establish several general methods for producing $\mathcal{D}^L$-cospectral graphs that can be used to construct infinite families. We provide examples showing that various properties are not preserved by $\mathcal{D}^L$-cospectrality, including examples of $\mathcal{D}^L$-cospectral strongly regular and circulant graphs. We establi…
It is shown that an undirected graph $G$ is cospectral with the Hermitian adjacency matrix of a mixed graph $D$ obtained from a subgraph $H$ of $G$ by orienting some of its edges if and only if $H=G$ and $D$ is obtained from $G$ by a four-way switching operation; if $G$ is connected, this happens if and only if $λ_1(G)=λ_1(D)$. All mixed graphs of rank 2 are determined and this is used to classify which mixed graphs of rank 2 are cospectral with respect to their Hermitian adjacency matrix. Several families of mixed graphs are found that are determined by their Hermitian spectrum in the sense t…
In this article, we generate large families of non-isomorphic and signless Lalacian cospectral graphs using partial transpose on graphs. Our constructions are significantly powerful. More than $70\%$ of non-isomorphic signless-Laplacian cospectral graphs can be generated with partial transpose when number of vertices is $\le 8$. We have also produced numerous examples of non-isomorphic signless Laplacian cospectral graphs.
A mixed multigraph is obtained from an undirected multigraph by orienting a subset of its edges. In this paper, we study a new Hermitian matrix representation of mixed multigraphs, give an introduction to cospectral operations on mixed multigraphs, and characterize switching equivalent mixed multigraphs in terms of fundamental cycle basis. As an application, an upper bound of cospectral classes of mixed multigraphs with the same underlying graph is obtained.
A well--known fact in Spectral Graph Theory is the existence of pairs of isospectral nonisomorphic graphs (known as PINGS). The work of A.J. Schwenk (in 1973) and of C. Godsil and B. McKay (in 1982) shed some light on the explanation of the presence of isospectral graphs, and they gave routines to construct PINGS. Here, we consider the Godsil-McKay--type routines developed for graphs, whose adjacency matrices are $(0,1)$-matrices, to the level of signed graphs, whose adjacency matrices allow the presence of $-1$'s. We show that, with suitable adaption, such routines can be successfully ported …
A set of graphs are called cospectral if their adjacency matrices have the same characteristic polynomial. In this paper we introduce a simple method for constructing infinite families of cospectral regular graphs. The construction is valid for special cases of a property introduced by Schwenk. For the case of cubic (3-regular) graphs, computational results are given which show that the construction generates a large proportion of the cubic graphs, which are cospectral with another cubic graph.
Several researchers have recently explored various graph parameters that can or cannot be characterized by the spectrum of a matrix associated with a graph. In this paper we show that several NP-hard zero forcing numbers are not characterized by the spectra of several types of associated matrices with a graph. In particular, we consider standard zero forcing, positive semidefinite zero forcing, and skew zero forcing, and provide constructions of infinite families of pairs of cospectral graphs which have different values for these numbers. We explore several methods for obtaining these cospectr…
A threshold graph G on n vertices is defined by binary sequence of length n. In this paper we present an explicit formula for computing the characteristic polynomial of a threshold graph from its binary sequence. Applications include obtaining a formula for the determinant of adjacency matrix of a threshold graph and showing that no two nonisomorphic threshold graphs are cospectral.
We present enumeration results on the number of connected graphs up to 10 vertices for which there is at least one other graph with the same spectrum (a cospectral mate), or at least one other graph with the same Smith normal form (coinvariant mate) with respect to several matrices associated to a graph. The present data give some indication that possibly the Smith normal form of the distance Laplacian and the signless distance Laplacian matrices could be a finer invariant to distinguish graphs in cases where other algebraic invariants, such as those derived from the spectrum, fail. Finally, w…
We introduce a zeta function of digraphs that determines, and is determined by, the spectra of all linear combinations of the adjacency matrix, its transpose, the out-degree matrix, and the in-degree matrix. In particular, zeta-equivalence of graphs encompasses simultaneous cospectrality with respect to the adjacency, the Laplacian, the signless Laplacian, and the normalized Laplacian matrix, respectively. In addition, we express zeta-equivalence in terms of Markov chains and in terms of invasions where each edge is replaced by a fixed digraph. We finish with a method for constructing zeta-equ…
Using an exhaustive search on cubic graphs of order 16, we find a unique cospectral pair with different chromatic indexes. This example indicates that the chromatic index of a regular graph is not characterized by its spectrum, which answers a question recently posed in [O. Etesami, W. H. Haemers, On NP-hard graph properties characterized by the spectrum, Discrete Appl. Math., 285(2020)526-529]. We prove that any orthogonal matrix representing the similarity between the two adjacency matrices of the cospectral pair cannot be rational. This implies that the cospectral pair cannot be obtained us…
Let $G$ be an $n$-vertex graph with adjacency matrix $A$, and $W=[e,Ae,\ldots,A^{n-1}e]$ be the walk matrix of $G$, where $e$ is the all-one vector. In Wang [J. Combin. Theory, Ser. B, 122 (2017): 438-451], the author showed that any graph $G$ is uniquely determined by its generalized spectrum (DGS) whenever $2^{-\lfloor n/2 \rfloor}\det W$ is odd and square-free. In this paper, we introduce a large family of graphs $\mathcal{F}_n=\{$ $n$-vertex graphs $G\colon\, 2^{-\lfloor n/2 \rfloor}\det W =p^2b$ and rank$W=n-1$ over $\mathbb{Z}/p\mathbb{Z}\},$ where $b$ is odd and square-free, $p$ is an o…
The traditional adjacency matrix of a mixed graph is not symmetric in general, hence its eigenvalues may be not real. To overcome this obstacle, several authors have recently defined and studied various Hermitian adjacency matrices of digraphs or mixed graphs. In this work we unify previous work and offer a new perspective on the subject by introducing the concept of monographs. Moreover, we consider questions of cospectrality.
The \textit{Kite graph}, denoted by $Kite_{p,q}$ is obtained by appending a complete graph $K_{p}$ to a pendant vertex of a path $P_{q}$. In this paper, firstly we show that no two non-isomorphic kite graphs are cospectral w.r.t adjacency matrix. Let $G$ be a graph which is cospectral with $Kite_{p,q}$ and the clique number of $G$ is denoted by $w(G)$. Then, it is shown that $w(G)\geq p-2q+1$. Also, we prove that $Kite_{p,2}$ graphs are determined by their adjacency spectrum.
Let $n$ be any positive integer, the friendship graph $F_n$ consist of $n$ edge-disjoint triangles that all of them meeting in one vertex. A graph $G$ is called cospectral with a graph $H$ if their adjacency matrices have the same eigenvalues. Recently in [http://arxiv.org/pdf/1310.6529v1.pdf] it is proved that if $G$ is any graph cospectral with $F_n$ $(n\neq 16)$, then $G\cong F_n$. In this note, we give a proof of special case of the latter: Any connected graph cospectral with $F_n$ is isomorphic to $F_n$. Our proof is independent of ones given in [http://arxiv.org/pdf/1310.6529v1.pdf] and …
We introduce a switching operation, inspired by the Godsil-McKay switching, in order to obtain pairs of $G$-cospectral gain graphs, that are gain graphs cospectral with respect to every representation of the gain group $G$. For instance, for two signed graphs, this notion of cospectrality is equivalent to the cospectrality of their signed adjacency matrices together with the cospectrality of their underlying graphs. Moreover, we introduce another more flexible switching in order to obtain pairs of gain graphs cospectral with respect to some fixed unitary representation. Many existing notions o…
For any graph, we define a rank-1 operator on a bipartite tensor product space, with components associated to the set of vertices and edges respectively. We show that the partial traces of the operator are the Laplacian and the edge-Laplacian. This provides an interpretation of the von Neumann entropy of the (normalized)\ Laplacian as the amount of quantum entanglement between two systems corresponding to vertices and edges. In this framework, cospectral graphs correspond exactly to local unitarily equivalent pure states. Finally, we introduce the notion of coentropic graphs, that is, graphs w…
A mixed extension of a graph $G$ is a graph $H$ obtained from $G$ by replacing each vertex of $G$ by a clique or a coclique, whilst two vertices in $H$ corresponding to distinct vertices $x$ and $y$ of $G$ are adjacent whenever $x$ and $y$ are adjacent in $G$. If $G$ is the path $P_3$, then $H$ has at most three adjacency eigenvalues unequal to $0$ and $-1$. Recently, the first author classified the graphs with the mentioned eigenvalue property. Using this classification we investigate mixed extension of $P_3$ on being determined by the adjacency spectrum. We present several cospectral familie…
Given simple graphs $G_1$ and $G_2$, the neighbourhood corona of $G_1$ and $G_2$, denoted $G_1\star G_2$, is the graph obtained by taking one copy of $G_1$ and $|V(G_1)|$ copies of $G_2$, and joining the neighbours of the $i$th vertex of $G_1$ to every vertex in the $i$th copy of $G_2$. In this paper we determine the adjacency spectrum of $G_1 \star G_2$ for arbitrary $G_1$ and $G_2$, and the Laplacian spectrum and signless Laplacian spectrum of $G_1\star G_2$ for regular $G_1$ and arbitrary $G_2$, in terms of the corresponding spectrum of $G_1$ and $G_2$. The results on the adjacency and sign…
The $α$-Hermitian adjacency matrix $H_α$ of a mixed graph $X$ has been recently introduced. It is a generalization of the adjacency matrix of unoriented graphs. In this paper, we consider a special case of the complex number $α$. This enables us to define an incidence matrix of mixed graphs. Consequently, we define a generalization of line graphs as well as a generalization of the signless Laplacian adjacency matrix of graphs. We then study the spectral properties of the signless Laplacian adjacency matrix of a mixed graph. Lastly, we characterize when the signless Laplacian adjacency matrix o…
We investigate when a complete graph $K_n$ with some edges deleted is determined by its adjacency spectrum. It is shown to be the case if the deleted edges form a matching, a complete graph $K_m$ provided $m \leq n-2$, or a complete bipartite graph. If the edges of a path are deleted we prove that the graph is determined by its generalized spectrum (that is, the spectrum together with the spectrum of the complement). When at most five edges are deleted from $K_n$, there is just one pair of nonisomorphic cospectral graphs. We construct nonisomorphic cospectral graphs (with cospectral complement…
The distance matrix $\mathcal{D}(G)$ of a graph $G$ is the matrix containing the pairwise distances between vertices. The transmission of a vertex $v_i$ in $G$ is the sum of the distances from $v_i$ to all other vertices and $T(G)$ is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, $\mathcal{D}^{\mathcal {L}}(G)=I-T(G)^{-1/2}\mathcal{D}(G) T(G)^{-1/2}$, is introduced. This is analogous to the normalized Laplacian matrix, $\mathcal{L}(G)=I-D(G)^{-1/2}A(G)D(G)^{-1/2}$, where $D(G)$ is the diagonal matrix of degrees of the vertices of the grap…
In this paper, we investigate (signless) Laplacian spectral characterization of graphs with star components. Also, we prove that the join graph ${\small K_{n-α}-e\vee αK_{1}}$ is $DQS$ for $n-α>3$ and $α\neq3$, and for $α=3$, we show that its only $Q$-cospectral mate is $\bar{K_{1,3}\cup K_{2}\cup (n-6)K_{1}}$.
In this paper, we investigate various algebraic and graph theoretic properties of the distance matrix of a graph. Two graphs are $D$-cospectral if their distance matrices have the same spectrum. We construct infinite pairs of $D$-cospectral graphs with different diameter and different Wiener index. A graph is $k$-transmission-regular if its distance matrix has constant row sum equal to $k$. We establish tight upper and lower bounds for the row sum of a $k$-transmission-regular graph in terms of the number of vertices of the graph. Finally, we determine the Wiener index and its complexity for l…
We study the spectra of mixed graphs about its Hermitian adjacency matrix of the second kind (i.e. N-matrix) introduced by Mohar [1]. We extend some results and define one new Hermitian adjacency matrix, and the entry corresponding to an arc from $u$ to $v$ is equal to the $k$-th( or the third) root of unity, i.e. $ω = cos(2π/k) + \textbf{i} \ sin(2π/k), k {\geq} 3$; the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. In this paper, we characterize the cospectrality conditions for a mixed graph and its underlying graph. In section 4, we determine a sharp upper bound o…
In this paper, we introduce central vertex corona, central edge corona, and central edge neighborhood corona of graphs using central graph. Also, we determine their adjacency spectrum, Laplacian spectrum and signless Laplacian spectrum. From our results, it is possible to obtain infinitely many pairs of adjacency (respectively, Laplacian and signless Laplacian) cospectral graphs. As an application, we calculate the number of spanning trees and the Kirchhoff index of the resulting graphs.
This contribution gives an extensive study on spectra of mixed graphs via its Hermitian adjacency matrix of the second kind { ($N$-matrix for short)} introduced by Mohar \cite{0001}. This matrix is indexed by the vertices of the mixed graph, and the entry corresponding to an arc from $u$ to $v$ is equal to the sixth root of unity $ω=\frac{1+{\bf i}\sqrt{3}}{2}$ (and its symmetric entry is $\barω=\frac{1-{\bf i}\sqrt{3}}{2}$); the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. The main results of this paper include the following: {equivalent} conditions for a mixed gr…
In this paper, it is shown that the graph $T_4(p,q,r)$ is determined by its Laplacian spectrum and there are no two non-isomorphic such graphs which are cospectral with respect to adjacency spectrum.
Two graphs are cospectral if their respective adjacency matrices have the same multiset of eigenvalues, and generalized cospectral if they are cospectral and so are their complements. We study generalized cospectrality in relation to logical definability. We show that any pair of graphs that are elementary equivalent with respect to the three-variable counting first-order logic $C^3$ are generalized cospectral, and this is not the case with $C^2$, nor with any number of variables if we exclude counting quantifiers. Using this result we provide a new characterization of the well-known class of …
An oriented hypergraph is an oriented incidence structure that generalizes and unifies graph and hypergraph theoretic results by examining its locally signed graphic substructure. In this paper we obtain a combinatorial characterization of the coefficients of the characteristic polynomials of oriented hypergraphic Laplacian and adjacency matrices via a signed hypergraphic generalization of basic figures of graphs. Additionally, we provide bounds on the determinant and permanent of the Laplacian matrix, characterize the oriented hypergraphs in which the upper bound is sharp, and demonstrate tha…
Let $G$ be a finite non abelian group. The centralizer graph of $G$ is a simple undirected graph $Γ_{cent}(G)$, whose vertices are the proper centralizers of $G$ and two vertices are adjacent if and only if their cardinalities are identical {\rm\cite{omer}}. The complement of the centralizer graph is called the co-centralizer graph. In this paper, we investigate the adjacency and (signless) Laplacian spectra of centralizer and co-centralizer graphs of some classes of finite non-abelian groups and obtain some conditions on a group so that the centralizer and co-centralizer graphs are adjacency,…
