Related papers: Some Bounds on the Energy of Graphs with Self-Loop…

For a graph $G$, let $S(G)$ be the Seidel matrix of $G$ and $\te_1(G),...,\te_n(G)$ be the eigenvalues of $S(G)$. The Seidel energy of $G$ is defined as $|\te_1(G)|+...+|\te_n(G)|$. Willem Haemers conjectured that the Seidel energy of any…

In 1993 Hong asked what are the best bounds on the $k$'th largest eigenvalue $\lambda_{k}(G)$ of a graph $G$ of order $n$. This challenging question has never been tackled for any $2<k<n$. In the present paper tight bounds are obtained for…

For a simple graph $G$ with $n$ vertices, $m$ edges and signless Laplacian eigenvalues $q_{1} \geq q_{2} \geq \cdots \geq q_{n} \geq 0$, its the signless Laplacian energy $QE(G)$ is defined as $QE(G) = \sum_{i=1}^{n}|q_{i} - \bar{d} |$,…

Let $D$ be a simple digraph with eigenvalues $z_1,z_2,...,z_n$. The energy of $D$ is defined as $E(D)= \sum_{i=1}^n |Re(z_i)|$, is the real part of the eigenvalue $z_i$. In this paper a lower bound will be obtained for the spectral radius…

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…

The trace norm $\left\Vert G\right\Vert _{\ast}$ of a graph $G$ is the sum of its singular values, i.e., the absolute values of its eigenvalues. The norm $\left\Vert G\right\Vert _{\ast}$ has been intensively studied under the name of graph…

For a graph $G$, the generalized adjacency matrix $A_\alpha(G)$ is the convex combination of the diagonal matrix $D(G)$ and the adjacency matrix $A(G)$ and is defined as $A_\alpha(G)=\alpha D(G)+(1-\alpha) A(G)$ for $0\leq \alpha \leq 1$.…

The energy of a simple graph $G$ arising in chemical physics, denoted by $\mathcal E(G)$, is defined as the sum of the absolute values of eigenvalues of $G$. We consider the asymptotic energy per vertex (say asymptotic energy) for lattice…

A digraph with self-loops $D_S$ with vertex set $\mathcal{V}$ is a simple digraph with a self-loop attached at every vertex in $S \subset \mathcal{V}.$ In this paper, we study the energy $E(D_S)$ of $D_S$ and its properties, which extend…

The energy of a graph $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of $G$. Let $G$ be a graph of order $n$ and ${\rm rank}(G)$ be the rank of the adjacency matrix of $G$. In this paper we…

Let $G$ be a simple undirected graph with adjacency matrix $A(G)$. The energy of $G$ is defined as the sum of absolute values of all eigenvalues of $A(G)$, which was introduced by Gutman in 1970s. Since graph energy has important chemical…

Let $G$ be a graph with maximum degree $\Delta$, and let $G^{\sigma}$ be an oriented graph of $G$ with skew adjacency matrix $S(G^{\sigma})$. The skew spectral radius $\rho_s(G^{\sigma})$ of $G^\sigma$ is defined as the spectral radius of…

Let $G$ be a simple graph of order $n$ with eigenvalues $\lambda_1(G)\geq \cdots \geq \lambda_n(G)$. Define \[s^+(G)=\sum_{\lambda_i >0} \lambda_i^2(G), \quad s^-(G)=\sum_{\lambda_i<0} \lambda_i^2(G).\] It was conjectured by Elphick,…

Let $G$ be a digraph with adjacency matrix $A(G)$ and outdegrees diagonal matrix $D(G)$. For any real $\alpha\in[0,1]$, the $A_\alpha$ matrix $A_\alpha(G)$ of a digraph $G$ is defined as $A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G)$. The…

Let $G$ be a graph with the vertex set $ \lbrace v_1,\ldots,v_n \rbrace$. The Seidel matrix of $G$ is an $n\times n$ matrix whose diagonal entries are zero, $ij$-th entry is $-1$ if $ v_{i} $ and $ v_{j} $ are adjacent and otherwise is $ 1…

Let $G$ be a simple undirected graph and $G^\sigma$ be the corresponding oriented graph of $G$ with the orientation $\sigma$. The skew energy of $G^\sigma$, denoted by $\varepsilon_s(G^\sigma)$, is defined as the sum of the singular values…

The eccentricity matrix $\varepsilon(G)$ of a graph $G$ is obtained from the distance matrix by retaining the eccentricities (the largest distance) in each row and each column. In this paper, we give a characterization of the star graph,…

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…

The energy $E(G)$ of a graph $G$ is defined as the sum of the absolute values of its eigenvalues. Let $S_2$ be the star of order 2 (or $K_2$) and $Q$ be the graph obtained from $S_2$ by attaching two pendent edges to each of the end…

The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of G. It is proved that E(G)>= 2(n-\chi(\bar{G}))>= 2(ch(G)-1) for every graph G of order n, and that E(G)>= 2ch(G) for all graphs G…