Related papers: On the spectrum of complex unit gain graph

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For every real $\alpha\in\left[ 0,1\right],$ define the matrix $A_{\alpha}\left(G\right) $ as \[ A_{\alpha}\left(G\right) =\alpha…

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.…

The adjacency-diametrical matrix (AD matrix) of a connected graph $G$ with diameter $d$, denoted by $AD(G)$, is the matrix indexed by the vertices of $G$ in which the $(i,j)$-entry of $AD(G)$ is $1$ if $d_G(v_i,v_j)=1$, is $d$ if…

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…

We describe the structure of those graphs that have largest spectral radius in the class of all connected graphs with a given degree sequence. We show that in such a graph the degree sequence is non-increasing with respect to an ordering of…

Given a connected graph $R$ on $r$ vertices and a rooted graph $H,$ let $R\{H\}$ be the graph obtained from $r$ copies of $H$ and the graph $R$ by identifying the root of the $i-th$ copy of $H$ with the $i-th$ vertex of $R$. Let…

A spanning tree $T$ of a connected graph $G$ is a subgraph of $G$ that is a tree covers all vertices of $G$. The leaf distance of $T$ is defined as the minimum of distances between any two leaves of $T$. A fractional matching of a graph $G$…

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…

For a simple graph $G$, the generalized adjacency matrix $A_{\alpha}(G)$ is defined as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G), \alpha\in [0,1]$, where $A(G)$ is the adjacency matrix and $D(G)$ is the diagonal matrix of vertex degrees of…

Let $G$ be a graph with $n$ vertices, and let $A(G)$ and $D(G)$ denote respectively the adjacency matrix and the degree matrix of $G$. Define $$ A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G) $$ for any real $\alpha\in [0,1]$. The collection of…

The power graph $\mathscr{P}(G)$ of a group $G$ is an undirected graph with all the elements of $G$ as vertices and where any two vertices $u$ and $v$ are adjacent if and only if $u=v^m $ or $v=u^m$, $ m \in$ $\mathbb{Z}$. For a simple…

A complex unit gain graph is a triple $\varphi=(G, \mathbb{T}, \varphi)$ (or $G^{\varphi}$ for short) consisting of a simple graph $G$, as the underlying graph of $G^{\varphi}$, the set of unit complex numbers $\mathbb{T}={z\in \mathbb{C}:…

In this paper we will apply the tensor and its traces to investigate the spectral characterization of unicyclic graphs. Let $G$ be a graph and $G^m$ be the $m$-th power (hypergraph) of $G$. The spectrum of $G$ is referring to its adjacency…

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 $…

A theory of orientation on gain graphs (voltage graphs) is developed to generalize the notion of orientation on graphs and signed graphs. Using this orientation scheme, the line graph of a gain graph is studied. For a particular family of…

A complex unit gain graph (or a $\mathbb{T}$-gain graph) $\Theta(\Sigma,\varphi)$ is a graph where the unit complex number is assign by a function $\varphi$ to every oriented edge of $\Sigma$ and assign its inverse to the opposite…

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…

Given a collection of graphs $\mathbf{G}=(G_1, \ldots, G_m)$ with the same vertex set, an $m$-edge graph $H\subset \cup_{i\in [m]}G_i$ is a transversal if there is a bijection $\phi:E(H)\to [m]$ such that $e\in E(G_{\phi(e)})$ for each…

The arithmetic-geometric matrix $A_{ag}(G)$ of a graph $G$ is a square matrix, where the $(i,j)$-entry is equal to $\displaystyle \frac{d_{i}+d_{j}}{2\sqrt{d_{i}d_{j}}}$ if the vertices $v_{i}$ and $v_{j}$ are adjacent, and 0 otherwise. The…

In this paper, we introduce a matrix for a mixed graph, called the integrated adjacency matrix. This matrix uniquely determines a mixed graph, as long as the indices of the matrix are specified. Additionally, we associate an (undirected)…