Results on Graph Energy

Objectives : The absolute sum of the eigenvalues is the definition of the graph's energy. In addition to discussing their relationship, this study compares the energy of the Adjacency matrix, Laplacian matrix, Signless Laplacian matrix, and Seidel matrix applied to ten distinct kinds of graphs. In this study, a correlation is established between the energy of graphs and the energy of Edge Antimagic graphs. Methods: The technical description of the graph's energy, . An Edge Antimagic graph's energy, identified by (G) = , is the absolute total of its eigenvalues. Findings: The energy was calculated along with its link to the energy of Edge Antimagic matrix, Adjacency matrix, Laplacian matrix, Signless Laplacian matrix, and Seidel matrix. Novelty: Edge antimagic graphs have been applied using the concept of energy. The energy of Edge Antimagic graphs and the energy of graphs were discovered to be related. An analysis was conducted comparing Adjacency energy, Seidel energy, Laplacian energy, and Signless Laplacian energy. Keywords: Laplacian Energy, Spectrum, Seidel Matrix, Signless Laplacian Matrix, Edge Antimagic Graphs.

Relation between signless Laplacian energy, energy of graph and its line graph

Integral circulant graphs of prime power order with maximal energy

An edge grafting theorem on the energy of unicyclic and bipartite graphs

A molecular graph is a graph in which vertices are atoms and edges are molecular bonds. These graphs are good mathematical models for molecules. Suppose G is a molecular graph with adjacency matrix A. The graph energy G is defined as the sum of the absolute values of the eigenvalues of A. The aim of this chapter is to describe a method for computing energy of fullerenes. We apply this method for computing a lower bound for energy of an infinite class of fullerene graphs with exactly 12n vertices. Our method is general and can be extended to other class of fullerene graphs.

After introducing and developing neutrosophic set theory, several research has been conducted in this area. In this chapter, the concept of energy of fuzzy graph, intuitionistic fuzzy graph, single valued neutrosophic graph are extended to the energy of a complex neutrosophic graph. We have defined the adjacency matrix of a complex neutrosophic graph and the energy of complex neutrosophic graph is defined in terms of its adjacency matrix. The lower and upper bounds for the energy of complex neutrosophic graph are also derived. In addition, we have defined the concept of Laplacian energy of a complex neutrosophic graph. A numerical example is provided for these concepts.

Introduction/purpose: The paper presents numerous vertex-degree-based graph invariants considered in the literature. A matrix can be associated to each of these invariants. By means of these matrices, the respective vertex-degree-based graph energies are defined as the sum of the absolute values of the eigenvalues. Results: The article determines the conditions under which the considered graph energies are greater or smaller than the ordinary graph energy (based on the adjacency matrix). Conclusion: The results of the paper contribute to the theory of graph energies as well as to the theory of vertex-degree-based graph invariants.

A method for computing local contributions to graph energy based on Estrada–Benzi approach

Unicyclic graphs with given number of pendent vertices and minimal energy

Robustness and Optimization of Complex Networks: Reconstructability, Algorithms and Modeling

Solution to a conjecture on the maximal energy of bipartite bicyclic graphs

Let GX be a graph obtained from a simple graph G by attaching a self-loop at each vertex of X ⊆ V ( G ) . The general extended adjacency matrix for the graph GX is defined and the bounds for the degree based energy of the graph GX are obtained. The study extends the notion of degree based energy of simple graphs to graphs with self-loops. For the graph GX of order n and size m with σ self-loops, the adjacency energy, E ( G X ) ≥ 4 mn + 2 σ ( n − σ ) n . The spectral radius ρ ( G X ) of its adjacency matrix is always less than or equal to 1 2 [ − 1 + 1 + 4 ( 2 m + σ + Δ − 1 ) ] , where Δ is the maximum degree in the graph GX and the equality conditions are given for 0 < σ < n . Few more bounds for ρ ( G X ) are also obtained. The study shows that, the spectral radius f max ( G X ) of its extended adjacency matrix satisfies: min i ∼ j F ij ( 2 m + σ ) n ≤ f max ( G X ) ≤ max i ∼ j F ij 2 ( 2 m + σ ) . We conclude the article by computing the extended adjacency spectrum of complete graph and complete bipartite graphs with self-loops.

Let E(G) be the energy of a graph G, which is defined as the sum of the absolute values of all eigenvalues of G. If G contains no cycle , then it is called a triangle-free graph. In this paper, we investigate the upper bounds of the energy for triangle-free graphs in terms of their matching number and characterize all the extremal graphs attaining the upper bounds. In addition, we establish a relation between graph energy and rank.

The energy of a graph is the sum of the absolute values of its eigenvalues. We propose a new problem on graph energy change due to any single edge deletion. Then we survey the literature for existing partial solution of the problem, and mention a conjecture based on numerical evidence. Moreover, we prove in three different ways that the energy of a cycle graph decreases when an arbitrary edge is deleted except for the order of 4.

Graph energy is defined to be the p-norm of adjacency matrix associated to the graph for p = 1 elaborated as the sum of the absolute eigenvalues of adjacency matrix. The graph’s spectral radius represents the adjacency matrix’s largest absolute eigenvalue. Applications for graph energies and spectral radii can be found in both molecular computing and computer science. On similar lines, Inverse Sum Indeg, (ISI) energies, and (ISI) spectral radii can be constructed. This article’s main focus is the ISI energies, and ISI spectral radii of the generalized splitting and shadow graphs constructed on any regular graph. These graphs can be representation of many physical models like networks, molecules and macromolecules, chains or channels. We actually compute the relations about the ISI energies and ISI spectral radii of the newly created graphs to those of the original graph.