Related papers: On the eccentricity energy of complete mutipartite…

For a connected graph $G$, we present the concept of a new graph matrix related to its distance and Seidel matrix, called distance Seidel matrix $\mathcal{D}^S(G)$. Suppose that the eigenvalues of $\mathcal{D}^S(G)$ be $\partial_{1}^{S}(G)…

Let $G$ be a connected graph of order $n$. The eccentricity $e(v)$ of a vertex $v$ is the distance from $v$ to a vertex farthest from $v$. The average eccentricity of $G$ is the mean of all eccentricities in $G$. We give upper bounds on the…

The \emph{eccentricity} of a vertex $u$ in a graph $G$, denoted by $e_G(u)$, is the maximum distance from $u$ to other vertices in $G$. We study extremal problems for the average eccentricity and the first and second Zagreb eccentricity…

The eccentric connectivity index of a graph $G$, denoted by $\xi^{c}(G)$, defined as $\xi^{c}(G)$ = $\sum_{v \in V(G)}\epsilon(v) \cdot d(v)$, where $\epsilon(v)$ and $d(v)$ denotes the eccentricity and degree of a vertex $v$ in a graph…

For a simple connected graph $G$ of order $n$ having distance Laplacian eigenvalues $ \rho^{L}_{1}\geq \rho^{L}_{2}\geq \cdots \geq \rho^{L}_{n}$, the distance Laplacian energy $DLE(G)$ is defined as…

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…

In his survey "Beyond graph energy: Norms of graphs and matrices" (2016), Nikiforov proposed two problems concerning characterizing the graphs that attain equality in a lower bound and in a upper bound for the energy of a graph,…

For a graph $G$ with vertex set $V(G)=\{v_1, v_2, \cdots, v_n\}$, the extended double cover $G^*$ is a bipartite graph with bipartition (X, Y), $X=\{x_1, x_2, \cdots, x_n\}$ and $Y=\{y_1, y_2, \cdots, y_n\}$, where two vertices $x_i$ and…

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…

Let G be a non-abelian group and Z(G) be the center of G. The non-commuting graph {\Gamma}(G) of G is a graph with vertex set is non central elements of G and two vertices x, y are adjacent if and only if they are commute. In this paper we…

Let $G$ be a connected edge-weighted graph of order $n$ and size $m$. Let $w:E(G)\rightarrow \mathbb{R}^{\geq 0}$ be the weighting function. We assume that $w$ is normalised, that is, $\sum_{e\in E(G)} w(e)=m$. The weighted distance…

The eccentric connectivity index of a connected graph $G$ is the sum over all vertices $v$ of the product $d_{G}(v) e_{G}(v)$, where $d_{G}(v)$ is the degree of $v$ in $G$ and $e_{G}(v)$ is the maximum distance between $v$ and any other…

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…

We study the extremal energy problem for complex unit gain graphs whose underlying graph is the dumbbell graph $D_{r,s,\ell}$. Using switching equivalence, we reduce the spectrum to the real parts of the two cycle gains and obtain an…

Given any graph $G$, the (adjacency) spread of $G$ is the maximum absolute difference between any two eigenvalues of the adjacency matrix of $G$. In this paper, we resolve a pair of 20-year-old conjectures of Gregory, Hershkowitz, and…

Let $G$ be a graph of order $n$, and let $A(G)$ and $D(G)$ be the adjacency matrix and the degree matrix of $G$ respectively. Define the convex linear combinations $A_\alpha (G)$ of $A (G)$ and $D (G) $ by $$A_\alpha (G)=\alpha…

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}…

It is known that complete multipartite graphs are determined by their distance spectrum but not by their adjacency spectrum. The Seidel spectrum of a graph $G$ on more than one vertex does not determine the graph, since any graph obtained…

Let G be an undirected graph on n vertices and let S(G) be the set of all real symmetric n x n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The inverse inertia problem for G…

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…