Let $0 \in Γ$ and $Γ\setminus \{0\}$ be a subgroup of the complex numbers of unit modulus. Define $\mathcal{H}_{n}(Γ)$ to be the set of all $n\times n$ Hermitian matrices with entries in $Γ$, whose diagonal entries are zero. The matrices $A,B\in \mathcal{H}_{n}(Γ)$ are said to be switching equivalent if there is a diagonal matrix $D$, in which the diagonal entries belong to $Γ\setminus \{0\}$, such that $D^{-1} A D=B$. We find a characterization, in terms of fundamental cycles of graphs, of switching equivalence of matrices in $\mathcal{H}_{n}(Γ)$. We give sufficient conditions to characterize the cospectral matrices in $\mathcal{H}_{n}(Γ)$. We find bounds on the number of switching equivalence classes of all mixed graphs with the same underlying graph. We also provide the size of all switching equivalence classes of mixed cycles, and give a formula that calculates the size of a switching equivalence class of a mixed plane graph. We also discuss an action of the automorphism group of a graph on switching equivalence classes of matrices in $\mathcal{H}_{n}(Γ)$.
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.
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…
We study the equivalence between bipartiteness and symmetry of spectra of mixed graphs, for $θ$-Hermitian adjacency matrices defined by an angle $θ\in (0, π]$. We show that this equivalence holds when, for example, an angle $θ$ is an algebraic number, while it breaks down for any angle $θ\in \mathbb{Q}π$. Furthermore, we construct a family of non-bipartite mixed graphs having the symmetric spectra for given $θ\in \mathbb{Q}π$.
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…
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.
In an earlier work, the author together with Guo [Hermitian adjacency matrix of digraphs and mixed graphs, J. Graph Theory 85 (2017) 217-248] introduced the Hermitian adjacency matrix of directed (and partially directed) graphs. However, it appears that a more natural Hermitian matrix exists, and it is the purpose of this note to bring this new Hermitian matrix to the attention of researchers in algebraic graph theory.
Let $X$ be bipartite mixed graph and for a unit complex number $α$, $H_α$ be its $α$-hermitian adjacency matrix. If $X$ has a unique perfect matching, then $H_α$ has a hermitian inverse $H_α^{-1}$. In this paper we give a full description of the entries of $H_α^{-1}$ in terms of the paths between the vertices. Furthermore, for $α$ equals the primitive third root of unity $γ$ and for a unicyclic bipartite graph $X$ with unique perfect matching, we characterize when $H_γ^{-1}$ is $\pm 1$ diagonally similar to $γ$-hermitian adjacency matrix of a mixed graph. Through our work, we have provided a n…
The paper gives a thorough introduction to spectra of digraphs via its Hermitian adjacency matrix. This matrix is indexed by the vertices of the digraph, and the entry corresponding to an arc from $x$ to $y$ is equal to the complex unity $i$ (and its symmetric entry is $-i$) if the reverse arc $yx$ is not present. We also allow arcs in both directions and unoriented edges, in which case we use $1$ as the entry. This allows to use the definition also for mixed graphs. This matrix has many nice properties; it has real eigenvalues and the interlacing theorem holds for a digraph and its induced su…
We propose a quantum walk defined by digraphs (mixed graphs). This is like Grover walk that is perturbed by a certain complex-valued function defined by digraphs. The discriminant of this quantum walk is a matrix that is a certain normalization of generalized Hermitian adjacency matrices. Furthermore, we give definitions of the positive and negative supports of the transfer matrix, and clarify explicit formulas of their supports of the square. In addition, we give tables by computer on the identification of digraphs by their eigenvalues.
Let $G$ be a simple undirected graph, and $G^ϕ$ be a mixed graph of $G$ with the generalized orientation $ϕ$ and Hermitian-adjacency matrix $H(G^ϕ)$. Then $G$ is called the underlying graph of $G^ϕ$. The Hermitian energy of the mixed graph $G^ϕ$, denoted by $\mathcal{E}_H(G^ϕ)$, is defined as the sum of all the singular values of $H(G^ϕ)$. A $k$-regular mixed graph on $n$ vertices having Hermitian energy $n\sqrt{k}$ is called a $k$-regular optimum Hermitian energy mixed graph. In this paper, we first focus on the problem proposed by Liu and Li [J. Liu, X. Li, Hermitian-adjacency matrices and H…
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…
Godsil-McKay switching is an operation on graphs that doesn't change the spectrum of the adjacency matrix. Usually (but not always) the obtained graph is non-isomorphic with the original graph. We present a straightforward sufficient condition for being isomorphic after switching, and give examples which show that this condition is not necessary. For some graph products we obtain sufficient conditions for being non-isomorphic after switching. As an example we find that the tensor product of the $\ell\times m$ grid ($\ell>m\geq 2$) and a graph with at least one vertex of degree two is not deter…
Let $M$ be a mixed graph and $H(M)$ be its Hermitian-adjacency matrix. If we add every edge and arc in $M$ a Randić weight, then we can get a new weighted Hermitian-adjacency matrix. What are the properties of this new matrix? Motivated by this, we define the Hermitian-Randić matrix $R_{H}(M)=(r_{h})_{kl}$ of a mixed graph $M$, where $(r_{h})_{kl}=-(r_{h})_{lk}=\frac{\textbf{i}}{\sqrt{d_{k}d_{l}}}$ ($\textbf{i}=\sqrt{-1}$) if $(v_{k},v_{l})$ is an arc of $M$, $(r_{h})_{kl}=(r_{h})_{lk}=\frac{1}{\sqrt{d_{k}d_{l}}}$ if $v_{k}v_{l}$ is an undirected edge of $M$, and $(r_{h})_{kl}=0$ otherwise. In…
A pointed graph $(Γ,v_0)$ induces a family of transition matrices in Wildberger's construction of a hermitian hypergroup via a random walk on $Γ$ starting from $v_0$. We will give a necessary condition for producing a hermitian hypergroup as we assume a weaker condition than the distance-regularity for $(Γ,v_0)$. The condition obtained in this paper connects the transition matrices and the adjacency matrices associated with $Γ$.
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…
A mixed graph is called \emph{second kind hermitian integral}(or \emph{HS-integral}) if the eigenvalues of its Hermitian-adjacency matrix of second kind are integers. A mixed graph is called \emph{Eisenstein integral} if the eigenvalues of its (0, 1)-adjacency matrix are Eisenstein integers. Let $Γ$ be an abelian group. We characterize the set $S$ for which a mixed Cayley graph $\text{Cay}(Γ, S)$ is HS-integral. We also show that a mixed Cayley graph is Eisenstein integral if and only if it is HS-integral.
It is proved that for any finite connected graph $G$, there exists an orientation of $G$ such that the spectral radius of the corresponding Hermitian adjacency matrix is smaller or equal to the spectral radius of the universal cover of $G$ (with equality if and only if $G$ is a tree). This resolves a problem proposed by Mohar. The proof uses the method of interlacing families of polynomials that was developed by Marcus, Spielman, and Srivastava in their seminal work on the existence of infinite families of Ramanujan graphs.
A mixed graph is called \emph{second kind hermitian integral}(or \emph{HS-integral}) if the eigenvalues of its Hermitian-adjacency matrix of second kind are integers. A mixed graph is called \emph{Eisenstein integral} if the eigenvalues of its (0, 1)-adjacency matrix are Eisenstein integers. We characterize the set $S$ for which a mixed circulant graph $\text{Circ}(\mathbb{Z}_n, S)$ is HS-integral. We also show that a mixed circulant graph is Eisenstein integral if and only if it is HS-integral. Further, the eigenvalues and the HS-eigenvalues of some oriented circulant graphs are expressed in …
The Hoffman program with respect to any real or complex square matrix $M$ associated to a graph $G$ stems from A. J. Hoffman's pioneering work on the limit points for the spectral radius of adjacency matrices of graphs less than $\sqrt{2+\sqrt{5}}$. The program consists of two aspects: finding all the possible limit points of $M$-spectral radii of graphs and detecting all the connected graphs whose $M$-spectral radius does not exceed a fixed limit point. In this paper, we summarize the results on this topic concerning several graph matrices, including the adjacency, the Laplacian, the signless…
It is customary to identify sparse matrices with the corresponding adjacency or incidence graph. For the solution of linear systems of equations using Gaussian elimination, the representation by its adjacency graph allows a symbolic computation that can be used to predict memory footprints and enables the determination of near-optimal elimination orderings based on heuristics. The Hermitian eigenvalue problem on the other hand seems to evade such treatment at first glance due to its inherent iterative nature. In this paper we prove this assertion wrong by showing the equivalence of the Hermiti…
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…
We study the behaviour of the 2-rank of the adjacency matrix of a graph under Seidel and Godsil-McKay switching, and apply the result to graphs coming from graphical Hadamard matrices of order $4^m$. Starting with graphs from known Hadamard matrices of order $64$, we find (by computer) many Godsil-McKay switching sets that increase the 2-rank. Thus we find strongly regular graphs with parameters $(63,32,16,16)$, $(64,36,20,20)$, and $(64,28,12,12)$ for almost all feasible 2-ranks. In addition we work out the behaviour of the 2-rank for a graph product related to the Kronecker product for Hadam…
It is known that complete multipartite graphs are determined by their distance spectrum but not by their adjacency spectrum. The Seidel spectrum of a graph $G$ on more than one vertex does not determine the graph, since any graph obtained from $G$ by Seidel switching has the same Seidel spectrum. We consider $G$ to be determined by its Seidel spectrum, up to switching, if any graph with the same spectrum is switching equivalent to a graph isomorphic to $G$. It is shown that any graph which has the same spectrum as a complete $k$-partite graph is switching equivalent to a complete $k$-partite g…
In this paper, we simplify the known switching theorem due to Bose and Shrikhande as follows. Let $G=(V,E)$ be a primitive strongly regular graph with parameters $(v,k,λ,μ)$. Let $S(G,H)$ be the graph from $G$ by switching with respect to a nonempty $H\subset V$. Suppose $v=2(k-θ_1)$ where $θ_1$ is the nontrivial positive eigenvalue of the $(0,1)$ adjacency matrix of $G$. This strongly regular graph is associated with a regular two-graph. Then, $S(G,H)$ is a strongly regular graph with the same parameters if and only if the subgraph induced by $H$ is $k-\frac{v-h}{2}$ regular. Moreover, $S(G,H…
A mixed graph is said to be HS-\emph{integral} if the eigenvalues of its Hermitian-adjacency matrix of the second kind are integers. A mixed graph is called \emph{Eisenstein integral} if the eigenvalues of its (0, 1)-adjacency matrix are Eisenstein integers. We characterize the set $S$ for which the normal mixed Cayley graph $\text{Cay}(Γ, S)$ is HS-integral for any finite group $Γ$. We further show that a normal mixed Cayley graph is HS-integral if and only if it is Eisenstein integral. This paper generalizes the results of [M. Kadyan, B. Bhattacharjya. HS-integral and Eisenstein integral mix…
A mixed graph is obtained from a graph by orienting some of its edges. The Hermitian adjacency matrix of a mixed graph with the vertex set $ \{v_{1}, \ldots , v_{n}\} $, is the matrix $ H=[h_{ij}]_{n \times n} $, where $ h_{ij}=-h_{ji}=i $ if there is a directed edge from $ v_{i} $ to $ v_{j} $, $ h_{ij}=1 $ if there exists an undirected edge between $v_i$ and $v_{j}$, and $h_{ij}=0$ otherwise. The Hermitian spectrum of a mixed graph is defined to be the spectrum of its Hermitian adjacency matrix. In this paper we study mixed graphs which are determined by their Hermitian spectrum (DHS). First…
We apply Godsil-McKay switching to the symplectic graphs over $\mathbb{F}_2$ with at least 63 vertices and prove that the 2-rank of (the adjacency matrix of) the graph increases after switching. This shows that the switched graph is a new strongly regular graph with parameters $(2^{2ν}\!-1, 2^{2ν-1}, 2^{2ν-2},2^{2ν-2})$ and 2-rank $2ν+2$ when $ν\geq 3$. For the symplectic graph on $63$ vertices we investigate repeated switching by computer and find many new strongly regular graphs with the above parameters for $ν=3$ with various 2-ranks. Using these results and a recursive construction method …
In this paper, we determine periodicity of quantum walks defined by mixed paths and mixed cycles. By the spectral mapping theorem of quantum walks, consideration of periodicity is reduced to eigenvalue analysis of $η$-Hermitian adjacency matrices. First, we investigate coefficients of the characteristic polynomials of $η$-Hermitian adjacency matrices. We show that the characteristic polynomials of mixed trees and their underlying graphs are same. We also define $n+1$ types of mixed cycles and show that every mixed cycle is switching equivalent to one of them. We use these results to discuss pe…
If a graph has a non-singular adjacency matrix, then one may use the inverse matrix to define a (labeled) graph that may be considered to be the inverse graph to the original one. It has been known that an adjacency matrix of a tree is non-singular if and only if the tree has a unique perfect matching; in this case the determinant of the matrix turns out to be $\pm 1$ and the inverse of the tree was shown to be `switching-equivalent' to a simple graph [C. Godsil, Inverses of Trees, Combinatorica 5 (1985), 33--39]. Using generalized inverses of symmetric matrices (that coincide with Moore-Penro…
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 …
We present the first steps towards the determination of the signed graphs for which the adjacency matrix has all but at most two eigenvalues equal to 1 or -1. Here we deal with the disconnected, the bipartite and the complete signed graphs. In addition, we present many examples which cannot be obtained from an unsigned graph or its negative by switching.
If all the eigenvalues of the Hermitian-adjacency matrix of a mixed graph are integers, then the mixed graph is called \emph{H-integral}. If all the eigenvalues of the (0,1)-adjacency matrix of a mixed graph are \emph{Gaussian integers}, then the mixed graph is called \emph{Gaussian integral}. For any finite group $Γ$, we characterize the set $S$ for which the normal mixed Cayley graph $\text{Cay}(Γ, S)$ is H-integral. We further prove that a normal mixed Cayley graph is H-integral if and only if it is Gaussian integral.
A mixed graph is said to be integral if all the eigenvalues of its Hermitian adjacency matrix are integer. Let $Γ$ be an abelian group. The \textit{mixed Cayley graph} $Cay(Γ,S)$ is a mixed graph on the vertex set $Γ$ and edge set $\{ (a,b): b-a\in S \}$, where $0\not\in S$. We characterize integral mixed Cayley graph $Cay(Γ,S)$ over abelian group in terms of its symbol set $S$.
A mixed circulant graph is called integral if all eigenvalues of its Hermitian adjacency matrix are integers. The main purpose of this paper is to investigate the existence of perfect state transfer (PST for short) and multiple state transfer (MST for short) on integral mixed circulant graphs. Concretely, we provide sufficient and necessary conditions for the existence of PST and MST between specified pairs of vertices on integral mixed circulant graphs, respectively.
Let $Γ$ denote an undirected, connected, regular graph with vertex set $X$, adjacency matrix $A$, and ${d+1}$ distinct eigenvalues. Let ${\mathcal A}={\mathcal A}(Γ)$ denote the subalgebra of Mat$_X({\mathbb C})$ generated by $A$. We refer to ${\mathcal A}$ as the {\it adjacency algebra} of $Γ$. In this paper we investigate algebraic and combinatorial structure of $Γ$ for which the adjacency algebra ${\mathcal A}$ is closed under Hadamard multiplication. In particular, under this simple assumption, we show the following: (i) ${\mathcal A}$ has a standard basis $\{I,F_1,\ldots,F_d\}$; (ii) for …
Let $G$ be a simple graph and $A(G)$ be the adjacency matrix of $G$. The matrix $S(G) = J -I -2A(G)$ is called the Seidel matrix of $G$, where $I$ is an identity matrix and $J$ is a square matrix all of whose entries are equal to 1. Clearly, if $G$ is a graph of order $n$ with no isolated vertex, then the Seidel matrix of $G$ is also the adjacency matrix of a signed complete graph $K_n$ whose negative edges induce $G$. In this paper, we study the Seidel eigenvalues of the complete multipartite graph $K_{n_1,\ldots,n_k}$ and investigate its Seidel characteristic polynomial. We show that if ther…
A continuous-time quantum walk on a graph $G$ is given by the unitary matrix $U(t) = \exp(-itA)$, where $A$ is the Hermitian adjacency matrix of $G$. We say $G$ has pretty good state transfer between vertices $a$ and $b$ if for any $ε> 0$, there is a time $t$, where the $(a,b)$-entry of $U(t)$ satisfies $|U(t)_{a,b}| \ge 1-ε$. This notion was introduced by Godsil (2011). The state transfer is perfect if the above holds for $ε= 0$. In this work, we study a natural extension of this notion called universal state transfer. Here, state transfer exists between every pair of vertices of the graph. W…
We study the balance of $G$-gain graphs, where $G$ is an arbitrary group, by investigating their adjacency matrices and their spectra. As a first step, we characterize switching equivalence and balance of gain graphs in terms of their adjacency matrices in $M_n(\mathbb C G)$. Then we introduce a represented adjacency matrix, associated with a gain graph and a group representation, by extending the theory of Fourier transforms from the group algebra $\mathbb C G$ to the algebra $M_n(\mathbb C G)$. We prove that a gain graph is balanced if and only if the spectrum of the represented adjacency ma…
A mixed graph is obtained by orienting some edges of a simple graph. The positive inertia index of a mixed graph is defined as the number of positive eigenvalues of its Hermitian adjacency matrix, including multiplicities. This matrix was introduced by Liu and Li, independently by Guo and Mohar, in the study of graph energy. Recently, Yuan et al. characterized the mixed graphs with exactly one positive eigenvalue. In this paper, we study the positive inertia indices of mixed graphs and characterize the mixed graphs with cut vertices having positive inertia index 2.
Let $G$ be a graph. For a subset $X$ of $V(G)$, the switching $σ$ of $G$ is the signed graph $G^σ$ obtained from $G$ by reversing the signs of all edges between $X$ and $V(G)\setminus X$. Let $A(G^σ)$ be the adjacency matrix of $G^σ$. An eigenvalue of $A(G^σ)$ is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. Let $S_{n,k}$ be the graph obtained from the complete graph $K_{n-r}$ by attaching $r$ pendent edges at some vertex of $K_{n-r}$. In this paper we prove that there exists a switching $σ$ such that all eigenvalues of $G^σ$ are main when $G$…
The spectral properties of signed directed graphs, which may be naturally obtained by assigning a sign to each edge of a directed graph, have received substantially less attention than those of their undirected and/or unsigned counterparts. To represent such signed directed graphs, we use a striking equivalence to $\mathbb{T}_6$-gain graphs to formulate a Hermitian adjacency matrix, whose entries are the unit Eisenstein integers $\exp(kπi/3),$ $k\in \mathbb{Z}_6.$ Many well-known results, such as (gain) switching and eigenvalue interlacing, naturally carry over to this paradigm. We show that n…
Let $G$ be a connected graph and let $T$ be a spanning tree of $G$. A partial orientation $σ$ of $G$ respect to $T$ is an orientation of the edges of $G$ except those edges of $T$, the resulting graph associated with which is denoted by $G_T^σ$. In this paper we prove that there exists a partial orientation $σ$ of $G$ respect to $T$ such that the largest eigenvalue of the Hermitian adjacency matrix of $G_T^σ$ is at most the largest absolute value of the roots of the matching polynomial of $G$.
Given a simple graph $G=(V_G, E_G)$ with vertex set $V_G$ and edge set $E_G$, the mixed graph $\widetilde{G}$ is obtained from $G$ by orienting some of its edges. Let $H(\widetilde{G})$ denote the Hermitian adjacency matrix of $\widetilde{G}$ and $A(G)$ be the adjacency matrix of $G$. The $H$-rank (resp. rank) of $\widetilde{G}$ (resp. $G$), written as $rk(\widetilde{G})$ (resp. $r(G)$), is the rank of $H(\widetilde{G})$ (resp. $A(G)$). Denote by $d(G)$ the dimension of cycle spaces of $G$, that is $d(G) = |E_G|-|V_G|+ω(G)$, where $ω(G),$ denotes the number of connected components of $G$. In t…
