![]() Topological indices and graph invariants based on vertex degree and the distance between vertices are commonly serve as indispensable tools in characterizing molecular graphs. Within the realm of chemical graph theory, degree-based topological indices assume a paramount role and hold immense significant. A topological index ( Gutman, 2013) is a numerical value intrinsically tied to a graph, serving as a fundamental characterizer of the graph’s topology while retaining its consistency through graph transformations. Notably, they have found applications especially in chemical disciplines ( Klein et al., 1992 Trinajstić and Nikolić, 2000), such as chemical documentation, isomer discrimination, study of molecular complexity, and other related fields like QSAR and QSPR, drug design, database choice, etc. In recent decades, topological indices have undergone extensive research in a number of fields, including mathematics ( Gutman and Polansky, 2012 Janezic et al., 2015), physics ( Labanowski et al., 1991), biology ( Bajorath and Bajorath, 2011). Moreover, A variety of topological indices have been built on the product of the degrees d k and d l of the terminal vertices k and l of the edge (chemical bond) kl, which has attracted the interest of mathematical chemists. Specifically in graph theory, the degree of a molecular graph vertex corresponds to the valency of an atom. There is a considerable and visible connection between chemistry and graph theory. They offer valuable insights into a graph’s connectivity, stability, expansion, and spread dynamics, making them indispensable for the analysis and characterization of graphs in different contexts. Both of these tools, i.e., the energy and spectral radius, play pivotal roles in comprehending the structural properties and behavior of graphs across various disciplines. Moreover, these mathematical tools have found several applications in algebraic graph theory ( Zhang et al., 2022). Notably, graph energy captures the graphical characteristic, while the spectral radius of a graph represents the largest absolute eigenvalue among all the eigenvalues of its adjacency matrix, denoted by the notation ℘( G). Consequently, there is a surge of interest in this field, leading to the constant emergence of graph energy and fundamental algebraic identities. At present, the idea of graph energy has gained substantial recognition due to its wide-ranging applications across diverse industries. Nevertheless, it wasn’t until the year 2000 that mathematicians truly embraced this concept. However, Initially, this notion was met with skepticism and was explored by only a limited group of scientists due to its unconventional nature. Furthermore, the concept of graph energy, denoted as ɛ( G), was initially introduced by Gutman in 1978 ( Gutman, 1978). Among these, the energy and spectral radius hold particular importance. Additionally, Graph theory encompasses various invariants that are significant for understanding the properties of a graph. ![]() The graph energy is one of the very few mathematical ideas which are chemically motivated into the modern subject of mathematics. With the help of these eigenvalues we define the spectrum of related graph G ( Cvetković et al., 2009) and then concluded their energy. The adjacency eigenvalues of G labelled as ( γ 1 ́, γ 2 ́, …, γ z ́ ) are the eigenvalues of A( G). The adjacency matrix for the graph G is a square matrix denoted as A( G) = Adj( G) =, in which = 1 when two vertices v k and v l are adjacent otherwise it defines to be as zero, mathematically it can be formulated as A ( G ) = 1 if v k ∼ v l, 0 elsewhere. We conventionally writes the vertex degree d k related to the vertex v k ∈ V( G) that represent total counting of edges end at a vertex v k of a graph. If v k and v l are two nearby vertices of graph G, then v k v l is used to refer to the edge that connects them. Let G be a simple undirected connected regular graph with the vertex set V( G) = and edge set E( G).
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