The Graph energy is the sum of absolute values of the eigenvalues of the (0; 1)-adjacency matrix. Oboudi recently obtained lower bounds for Graph energy, depending on the largest and smallest Graph eigenvalue. In this paper, a few more Oboudi-type bounds are deduced.

Let $G$ be a simple Graph with vertex set $V(G) = \{v_1, v_2,\ldots, v_n\}$. The elliptic Sombor matrix of $G$, denoted by $A_{ESO}(G)$, is defined as the $n\times n$ matrix whose $(i,j)$-entry is $(d_i+d_j)\sqrt{d_i^2+d_j^2}$ if $v_i$ and $v_j$ are adjacent and $0$ for another cases.Let the eigenvalues of the elliptic Sombor matrix $A_{ESO}(G)$ be $\rho_1\geq \rho_2\geq \ldots\geq \rho_n$ which are the roots of the elliptic Sombor characteristic polynomial $\prod_{i=1}^n (\rho-\rho_i)$. The elliptic Sombor energy ${E_{ESO}}$ of $G$ is the sum of absolute values of the eigenvalues of $A_{ESO}(G)$. In this paper, we compute the elliptic Sombor characteristic polynomial and the elliptic Sombor energy for some Graph classes. We compute the elliptic Sombor energy of cubic Graphs of order $10$ and as a consequence, we see that two $k$-regular Graphs of the same order may have different elliptic Sombor energy.

We introduce the Laplacian sum-eccentricity matrix LSe of a Graph G, and its Laplacian sum-eccentricity energy LSeE = Pn i=1 j ij, where  i =  i 􀀀 2m n and where  1;  2; : : :;  n are the eigenvalues of LSe. Upper bounds for LSeE are obtained. A Graph is said to be twinenergetic if Pn P i=1 j ij = n i=1 j ij. Conditions for the existence of such Graphs are established.

In this paper‎, ‎we introduce the concept of deficiency sum matrix $S_{df}(G)$ of a simple Graph $G=(V,E)$ of order $n$‎. ‎The deficiency $df(v)$ of a vertex $v \in V$ is the deviation between the degree of the vertex $v$‎‎ and the maximum degree of the Graph‎. ‎The deficiency sum matrix $S_{df}(G)$ is a matrix of order $n$ whose $(i,j)$-th entry is $df(v_{i})+df(v_{j})$‎, ‎if the vertices $v_{i}$ and $v_{j}$ are adjacent and $0$‎, ‎otherwise‎. ‎In addition‎, ‎we introduce deficiency sum energy $ES_{df}(G)$ of a Graph $G$ and establish some bounds for $ES_{df}(G)$‎. ‎Further‎, ‎deficiency sum energy of some classes of Graphs are obtained‎. ‎Moreover‎, ‎we construct an algorithm and python(3.8) code to find out spectrum and deficiency sum energy of Graph $G$‎.

Let G be a simple Graph with vertex set {v1, v2, ..., vn}. The common neighborhood Graph (conGraph) of G, denoted by con (G), is the Graph with vertex set {v1, v2, ..., vn}, in which two vertices are adjacent if and only they have at least one common neighbor in the Graph G. The basic properties of con (G) and of its energy are established.

The energy of a Graph G, denoted by E(G), is de ned as the sum of the absolute values of all eigenvalues of G. In this paper, lower and upper bounds for energy in some of the Graphs are established, in terms of Graph invariants such as the number of vertices, the number of edges, and the number of closed walks.

Let $G$ be a Graph with a vertex weight $omega$ and the vertices $v_1,ldots,v_n$. The Laplacian matrix of $G$ with respect to $omega$ is defined as $L_omega(G)=diag(omega(v_1),cdots,omega(v_n))-A(G)$, where $A(G)$ is the adjacency matrix of $G$. Let $mu_1,cdots,mu_n$ be eigenvalues of $L_omega(G)$. Then the Laplacian energy of $G$ with respect to $omega$ defined as $LE_omega (G)=sum_{i=1}^nbig|mu_i - overline{omega}big|$, where $overline{omega}$ is the average of $omega$, i.e., $overline{omega}=dfrac{sum_{i=1}^{n}omega(v_i)}{n}$. In this paper we consider several natural vertex weights of $G$ and obtain some inequalities between the ordinary and Laplacian energies of $G$ with corresponding vertex weights. Finally, we apply our results to the molecular Graph of toroidal fullerenes (or achiral polyhex nanotorus).\[5mm] noindenttextbf{Key words:} energy of Graph, Laplacian energy, Vertex weight, Topological index, toroidal fullerenes.

The energy of a Graph G is equal to the sum of absolute values of the eigenvalues of the adjacency matrix of G, whereas the Laplacian energy of a Graph G is equal to the sum of the absolute value of the difference between the eigenvalues of the Laplacian matrix of G and average degree of the vertices of G. Motivated by the work from Sharafdini and Panahbar [R. Sharafdini, H. Panahbar, Vertex weighted Laplacian Graph energy and other topological indices, J. Math. Nanosci. 6 (2016) 57 􀀀 65], in this paper we investigate the eccentricity version of Laplacian energy of a Graph G.