Related papers: The energy of random signed graph

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,…

The power graph $\Gamma_G$ of a finite group $G$ is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. In this paper, we classify the finite groups whose power graphs have…

Let $\Gamma=(G, \sigma)$ be a signed graph of order $n$ with underlying graph $G$ and a sign function $\sigma: E(G)\rightarrow \{+, -\}$. Denoted by $i_+(\Gamma)$, $\theta(\Gamma)$ and $p(\Gamma)$ the positive inertia index, the cyclomatic…

We study eigenvalue distribution of the adjacency matrix $A^{(N,p,q)}$ of weighted random uniform $q$-hypergraphs $\Gamma= \Gamma_{N,p,q}$. We assume that the graphs have $N$ vertices and the average number of hyperedges attached to one…

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,…

Signed graphs have their edges labeled either as positive or negative. $\rho(M)$ denote the $M$-spectral radius of $\Sigma$, where $M=M(\Sigma)$ is a real symmetric graph matrix of $\Sigma$. Obviously,…

We give a bound for the graph energy with given maximal degree in terms of the second and fourth moments of a graph. In the case in which the graph is $d$-regular we obtain the bound that is given in Van Dam, E. et al. (2014). through…

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…

In this paper we extend the concept of energy to signed digraphs. We obtain Coulson's integral formula for energy of signed digraphs. Formulae for energies of signed directed cycles are computed and it is shown that energy of non cycle…

The power graph of an arbitrary group $G$ is a simple graph with all elements of $G$ as its vertices and two vertices are adjacent if one is a positive power of another. In this paper, we generalize this concept to a graph whose vertices…

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…

Answering some questions of Gutman, we show that, except for four specific trees, every connected graph G of order n, with no cycle of order 4 and with maximum degree at most 3, has energy greater that its order. Here, the energy of a graph…

Circulant graphs are an important class of interconnection networks in parallel and distributed computing. Integral circulant graphs play an important role in modeling quantum spin networks supporting the perfect state transfer as well. The…

A signed graph $ (G, \Sigma)$ is a graph positive and negative ($\Sigma $ denotes the set of negative edges). To re-sign a vertex $v$ of a signed graph $ (G, \Sigma)$ is to switch the signs of the edges incident to $v$. If one can obtain $…

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 finite group, the enhanced power graph of $G$, denoted by $\Gamma_G^e$, is the graph with vertex set $G$ and two vertices $x,y$ are edge connected in $\Gamma_{G}^e$ if there exist $z\in G$ such that $x,y\in\langle z\rangle$.…

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…

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 this article we investigate normalized adjacency eigenvalues (simply normalized eigenvalues) and normalized adjacency energy of connected threshold graphs. A threshold graph can always be represented as a unique binary string. Certain…

In a signed graph each edge has a sign, $+1$ or $-1$. We introduce in the present paper a new definition of connection in a signed graph by the existence of both positive and negative chains between vertices. We prove some results and…