Related papers: More on Equienergetic Threshold Graphs

For a simple graph $G$ with adjacency matrix $A(G)$, let $\pi(G,x):=\mathrm{per}(xI-A(G))$ be its permanental polynomial with roots $\mu_1,\ldots,\mu_n \in \mathbb{C}$, and define the permanental energy $E_{\mathrm{per}}(G):=\sum_{i=1}^n…

The distance energy of a graph $G$ is a recently developed energy-type invariant, defined as the sum of absolute values of the eigenvalues of the distance matrix of $G$. There was a vast research for the pairs and families of non-cospectral…

Let $G$ be a graph on $n$ vertices with independence number $\alpha(G)$. Let $\mathcal{E}(G)$ be the energy of a graph, defined as the sum of the absolute values of the adjacency eigenvalues of $G$. Using Graffiti, Fajtlowicz conjectured in…

In this paper, we define and obtain several properties of the (adjacency) energy of a hypergraph. In particular, bounds for this energy are obtained as functions of structural and spectral parameters, such as Zagreb index and spectral…

For a graph $G$, we associate a family of real symmetric matrices, $S(G)$, where for any $A\in S(G)$, the location of the nonzero off-diagonal entries of $A$ are governed by the adjacency structure of $G$. Let $q(G)$ be the minimum number…

The total graph of $G$, $\mathcal T(G)$ is the graph whose set of vertices is the union of the sets of vertices and edges of $G$, where two vertices are adjacent if and only if they stand for either incident or adjacent elements in $G$. Let…

The GG-width of a class of graphs GG is defined as follows. A graph G has GG-width k if there are k independent sets N1,...,Nk in G such that G can be embedded into a graph H in GG such that for every edge e in H which is not an edge in G,…

Let $\eta_{1}\ge \eta_{2}\ge\cdots\ge \eta_{n}$ be the eigenavalues of $\mathcal{ABS}$ matrix. In this paper, we characterize connected graphs with $\mathcal{ABS}$ eigenvalue $\eta_{n}>-1$. As a result, we determine all connected graphs…

Let $\Gamma$ be a graph with the adjacency matrix $A$. The energy of $\Gamma$ is the sum of the absolute values of the eigenvalues of $A$. In this article we compute the energies of the commuting graphs of some finite groups and discuss…

In 1978 Gutman introduced the energy of a graph as the sum of the absolute values of graph eigenvalues, and ever since then graph energy has been intensively studied. Since graph energy is the trace norm of the adjacency matrix, matrix…

The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph. Akbari et al. \cite{S. Akbari} proved that for a complete multipartite graph $K_{t_1 ,\ldots,t_k}$, if $t_i\geq 2 \ (i=1,\ldots,k)$, then…

Given a graph $G$, let $G^\sigma$ be an oriented graph of $G$ with the orientation $\sigma$ and skew-adjacency matrix $S(G^\sigma)$. The skew energy of the oriented graph $G^\sigma$, denoted by $\mathcal{E}_S(G^\sigma)$, is defined as the…

For a given graph \( G \), let \( G^{(j)} \) denote the graph obtained by the deletion of vertex \( v_j \) from \( G \). The difference \( \mathscr{E}(G) - \mathscr{E}(G^{(j)}) \) quantifies the change in the energy of \( G \) upon the…

Let $G_S$ be a graph with loops obtained from a graph $G$ of order $n$ and loops at $S \subseteq V(G)$. In this paper, we establish a neccesary and sufficient condition on the bipartititeness of a connected graph $G$ and the spectrum…

We prove that, for any graph $G$, its graph energy is at least twice the Randic index. We show that equality holds if and only if $G$ is the union of complete bipartite graphs.

The eccentricity matrix $\varepsilon(G)$ of a graph $G$ is obtained from the distance matrix of $G$ by retaining the largest distances in each row and each column, and leaving zeros in the remaining ones. The eccentricity energy of $G$ is…

In this paper, we mainly study the trace norm of the adjacency matrix of a graph, also known as the energy of graph. We give the maximum trace norms for the graph and its complement. In fact, the above problem is stated and solved in a more…

Let $G$ be a simple graph with vertex set $V(G) = \{v_1, v_2,\ldots, v_n\}$. The Sombor matrix of $G$, denoted by $A_{SO}(G)$, is defined as the $n\times n$ matrix whose $(i,j)$-entry is $\sqrt{d_i^2+d_j^2}$ if $v_i$ and $v_j$ are adjacent…

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 distance characteristic polynomial of a threshold graph from its binary sequence. As application, we…

In this article we introduce a definition of k-uniform thresholds hypergraphs through a binary sequence, a natural extension of the classical definition for thresholds graphs. We characterize some of its eigenvalues and multiplicities by…