Related papers: L-Borderenergetic graphs

Let $G$ be a simple graph with order $n$ and size $m$. The quantity $M_1(G)=\displaystyle\sum_{i=1}^{n}d^2_{v_i}$ is called the first Zagreb index of $G$, where $d_{v_i}$ is the degree of vertex $v_i$, for all $i=1,2,\dots,n$. The signless…

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…

The sum of the absolute values of the eigenvalues of a graph is called the energy of the graph. We study the problem of finding graphs with extremal energy within specified classes of graphs. We develop tools for treating such problems and…

For a given simple graph $G$, the energy of $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let $P_n^{\ell}$ be the unicyclic graph obtained by connecting a vertex of $C_\ell$…

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,…

Denote the Laplacian of a graph $G$ by $L(G)$ and its second smallest Laplacian eigenvalue by $\lambda_2(G)$. If $G$ is a graph on $n\ge 2$ vertices, then it is shown that the second smallest eigenvalue of $L(G) + \frac{1}{n}…

In this paper, we introduce the notion of an energy graph as a simple, directed and vertex labeled graph $G$ such that the arcs $(v_i, v_j) \notin A(G)$ if $i > j$ for all distinct pairs $v_i,v_j$ and at least one vertex $v_k$ exists such…

We consider weighted graphs with an infinite set of vertices. We show that boundedness of all functions of finite energy can be seen as a notion of `relative compactness' for such graphs and study sufficient and necessary conditions for…

In this paper, we study energies associated with hypergraphs. More precisely, we obtain results for the incidence and the singless Laplacian energies of uniform hypergraphs. In particular, we obtain bounds for the incidence energy as…

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…

The matching energy is defined as the sum of the absolute values of the zeros of the matching polynomial of a graph, which is proposed first by Gutman and Wagner [The matching energy of a graph, Discrete Appl. Math. 160 (2012) 2177--2187].…

Let $H=(V,E)$ be an $r$-uniform hypergraph with the vertex set $V$ and the edge set $E$. For $1\leq s \leq r/2$, we define a weighted graph $G^{(s)}$ on the vertex set ${V\choose s}$ as follows. Every pair of $s$-sets $I$ and $J$ is…

The $k$-th Laplacian spectral moment of a digraph $G$ is defined as $\sum_{i=1}^n \lambda_i^k$, where $\lambda_i$ are the eigenvalues of the Laplacian matrix of $G$ and $k$ is a nonnegative integer. For $k=2$, this invariant is better known…

Let $G_{S}$ be a graph with $n$ vertices obtained from a simple graph $G$ by attaching one self-loop at each vertex in $S \subseteq V(G)$. The energy of $G_{S}$ is defined by Gutman et al. as $E(G_{S})=\sum_{i=1}^{n}\left| \lambda_{i}…

In this paper, we consider commuting conjugacy class graph (abbreviated as CCC-graph) of a finite group $G$ which is a graph with vertex set $\{x^G : x \in G \setminus Z(G)\}$ (where $x^G$ denotes the conjugacy class containing $x$) and two…

Gutman and Wagner proposed the concept of the matching energy (ME) and pointed out that the chemical applications of ME go back to the 1970s. Let $G$ be a simple graph of order $n$ and $\mu_1,\mu_2,\ldots,\mu_n$ be the roots of its matching…

Let $\Gamma$ be a simple graph with $n$ vertices. The energy of $\Gamma$, denoted by $\mathcal{E}(\Gamma)$, is defined as the sum of the absolute values of the eigenvalues of $\Gamma$. The graph $\Gamma$ is said to be hyperenergetic if…

In this paper we compute bounds for signless Laplacian energy, distance signless Laplacian eigenvalues and signless Laplacian energy of unitary addition Cayley graph G_{n}. We also obtain distance Laplacian eigenvalues and distance…

The energy $E$ of a graph is defined to be the sum of the absolute values of its eigenvalues. Nikiforov in {\it ``V. Nikiforov, The energy of $C_4$-free graphs of bounded degree, Lin. Algebra Appl. 428(2008), 2569--2573"} proposed two…

The concept of Randic index has been extended recently for a digraph. We prove that $2R(G)\leq \mathcal{E}(G)\leq 2\sqrt{\Delta(G)} R(G)$, where $G$ is a digraph, and $R(G)$ denotes the Randic index, $\mathcal{E}(G)$ denotes the Nikiforov…