相关论文: Hermitian-Randi\'c matrix and Hermitian-Randi\'c e…

Let $G$ be a simple undirected graph, and $G^\phi$ be a mixed graph of $G$ with the generalized orientation $\phi$ and Hermitian-adjacency matrix $H(G^\phi)$. Then $G$ is called the underlying graph of $G^\phi$. The Hermitian energy of the…

Let $X$ be a mixed graph and $\omega=\frac{1+\i \sqrt{3}}{2}$. We write $i\rightarrow j$, if there is an oriented edge from a vertex $v_i$ to another vertex $v_j$, and $i\sim j$ for an un-oriented edge between the vertices $v_i$ and $v_j$.…

Let $G$ be a simple graph with an orientation $\sigma$, which assigns to each edge a direction so that $G^\sigma$ becomes a directed graph. $G$ is said to be the underlying graph of the directed graph $G^\sigma$. In this paper, we define a…

Let $G$ be a simple graph with vertex set $V(G) = \{v_1, v_2,..., v_n\}$. The Randi\'{c} matrix of $G$, denoted by $R(G)$, is defined as the $n\times n$ matrix whose $(i,j)$-entry is $(d_id_j)^{\frac{-1}{2}}$ if $v_i$ and $v_j$ are adjacent…

Let $G$ be a simple graph with vertex set $V(G) = \{v_1, v_2,\ldots, v_n\}$ and edge set $E(G) = \{e_1, e_2,\ldots, e_m\}$. Similar to the Randi\'c matrix, here we introduce the Randi\'c incidence matrix of a graph $G$, denoted by $I_R(G)$,…

Let $G$ be a simple graph of order $n$. The energy $E(G)$ of the graph $G$ is the sum of the absolute values of the eigenvalues of $G$. The Randi\'{c} matrix of $G$, denoted by $R(G)$, is defined as the $n\times n$ matrix whose…

This paper explores the Harmonic matrix $MH(G)$ associated with a simple graph $ G $, where each entry corresponds to $ \frac{2}{d_i + d_j} $ for adjacent vertices $ v_i $ and $ v_j $. We investigate the spectral properties of this matrix,…

Let $G$ be a graph of order $n$, and $d_i$ the degree of a vertex $v_i$ of $G$. The Randi\'c matrix ${\bf R}=(r_{ij})$ of $G$ is defined by $r_{ij} = 1 / \sqrt{d_jd_j}$ if the vertices $v_i$ and $v_j$ are adjacent in $G$ and $r_{ij}=0$…

We give a new inequality between the energy of a graph and a weighted sum over the edges of the graph. Using this inequality we prove that $\mathcal{E}(G)\geq 2R(H)$, where $ \mathcal{E}(G)$ is the energy of a graph $G$ and $R(H)$ is the…

Let $R$ be a Hermitian matrix. The energy of $R$, $\mathcal{E}(R)$, corresponds to the sum of the absolute values of its eigenvalues. In this work it is obtained two lower bounds for $\mathcal{E}(R).$ The first one generalizes a lower bound…

For a Hermitian matrix $A$ of order $n$ with eigenvalues $\lambda_1(A)\ge \cdots\ge \lambda_n(A)$, define \[ \mathcal{E}_p^+(A)=\sum_{\lambda_i > 0} \lambda_i^p(A), \quad \mathcal{E}_p^-(A)=\sum_{\lambda_i<0} |\lambda_i(A)|^p,\] to be the…

In 2018, Arizmendi and Juarez introduced the concept of energy of a vertex, a novel approach allowing the total energy of a graph to be expressed as the sum of the energies of its individual vertices. In this article, we extend the notion…

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…

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 $…

In this paper, we determine the Randic' energy of the m-splitting graph, the m-shadow graph and the m-duplicate graph of a given graph, m being an arbitrary integer. Our results allow the construction of an infinite sequence of graphs…

Let $\varepsilon(G)$ be the eccentricity matrix of a graph $G$ and $Spec(\varepsilon(G))$ be the eccentricity spectrum of $G$. Let $H[G_1,G_2,\ldots, G_k]$ be the $H$-join of graphs $G_1,G_2,\ldots, G_k$ and let $H[G]$ be lexicographic…

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…

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…

The $\alpha$-Hermitian adjacency matrix $H_\alpha$ 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…

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…