Estimating vertex-degree-based energies

Introduction/purpose: In the current literature, several dozens of vertex-degree-based (VDB) graph invariants are being studied. To each such invariant, a matrix can be associated. The VDB energy is the energy (= sum of the absolute values of the eigenvalues) of the respective VDB matrix. The paper examines some general properties of the VDB energy of bipartite graphs. Results: Estimates (lower and upper bounds) are established for the VDB energy of bipartite graphs in which there are no cycles of size divisible by 4, in terms of ordinary graph energy. Conclusion: The results of the paper contribute to the spectral theory of VDB matrices, especially to the general theory of VDB energy.

The Minimum Edge Covering Energy of a Graph

In this paper, we introduce a new kind of graph energy, the minimum edge covering energy, ECE(G). It depends both on the underlying graph G, and on its particular minimum edge covering CE. Upper and lower bounds for ECE(G) are established. The minimum edge covering energy and some of the coefficients of the polynomial of well-known families of graphs like Star, Path and Cycle Graphs are computed

Graph energy and Hadamard's inequality

We use Hadamard's determinantal inequality and its generalization to prove some upper bounds on the energy of a graph in terms of degrees, average 2-degrees and number of common neighbors of its vertices. Also, we prove an inequality relating the energy of a graph and one arbitrary subgraph of it.

{0,1}-Brauer Configuration Algebras and Their Applications in Graph Energy Theory

The energy E(G) of a graph G is the sum of the absolute values of its adjacency matrix. In contrast, the trace norm of a digraph Q, which is the sum of the singular values of the corresponding adjacency matrix, is the oriented version of the energy of a graph. It is worth pointing out that one of the main problems in this theory consists of determining appropriated bounds of these types of energies for significant classes of graphs, digraphs and matrices, provided that, in general, finding out their exact values is a problem of great difficulty. In this paper, the trace norm of a {0,1}-Brauer configuration is introduced. It is estimated and computed by associating suitable families of graphs and posets to Brauer configuration algebras.

The minimum vertex–vertex dominating Laplacian energy of a graph

Let [Formula: see text] denote the set of all blocks of a graph [Formula: see text]. Two vertices are said to vv-dominate each other if they are vertices of the same block. A set [Formula: see text] is said to be vertex–vertex dominating set (vv-dominating set) if every vertex in [Formula: see text] is vv-dominated by some vertex in [Formula: see text]. The vv-domination number [Formula: see text] is the cardinality of the minimum vv-dominating set of [Formula: see text]. In this paper, we introduce new kind of graph energy, the minimum vv-dominating Laplacian energy of a graph denoting it as LE[Formula: see text]. It depends both on the underlying graph of [Formula: see text] and the particular minimum vv-dominating set of [Formula: see text]. Upper and lower bounds for LE[Formula: see text] are established and we also obtain the minimum vv-dominating Laplacian energy of some family of graphs.

Effect of a Ring onto Values of Eigenvalue–Based Molecular Descriptors

The eigenvalues of the characteristic polynomial of a graph are sensitive to its symmetry-related characteristics. Within this study, we have examined three eigenvalue–based molecular descriptors. These topological molecular descriptors, among others, are gathering information on the symmetry of a molecular graph. Furthermore, they are being ordinarily employed for predicting physico–chemical properties and/or biological activities of molecules. It has been shown that these indices describe well molecular features that are depending on fine structural details. Therefore, revealing the impact of structural details on the values of the eigenvalue–based topological indices should give a hunch how physico–chemical properties depend on them as well. Here, an effect of a ring in a molecule on the values of the graph energy, Estrada index and the resolvent energy of a graph is examined.

Novel Concept of Energy in Bipolar Single-Valued Neutrosophic Graphs with Applications

The energy of a graph is defined as the sum of the absolute values of its eigenvalues. Recently, there has been a lot of interest in graph energy research. Previous literature has suggested integrating energy, Laplacian energy, and signless Laplacian energy with single-valued neutrosophic graphs (SVNGs). This integration is used to solve problems that are characterized by indeterminate and inconsistent information. However, when the information is endowed with both positive and negative uncertainty, then bipolar single-valued neutrosophic sets (BSVNs) constitute an appropriate knowledge representation of this framework. A BSVNs is a generalized bipolar fuzzy structure that deals with positive and negative uncertainty in real-life problems with a larger domain. In contrast to the previous study, which directly used truth and indeterminate and false membership, this paper proposes integrating energy, Laplacian energy, and signless Laplacian energy with BSVNs to graph structure considering the positive and negative membership degree to greatly improve decisions in certain problems. Moreover, this paper intends to elaborate on characteristics of eigenvalues, upper and lower bound of energy, Laplacian energy, and signless Laplacian energy. We introduced the concept of a bipolar single-valued neutrosophic graph (BSVNG) for an energy graph and discussed its relevant ideas with the help of examples. Furthermore, the significance of using bipolar concepts over non-bipolar concepts is compared numerically. Finally, the application of energy, Laplacian energy, and signless Laplacian energy in BSVNG are demonstrated in selecting renewable energy sources, while optimal selection is suggested to illustrate the proposed method. This indicates the usefulness and practicality of this proposed approach in real life.