Let \(E(G)\) and \(d_x\) denote the edge set and degree of a vertex \(x\) in \(G\), respectively. Recently, the elliptic Sombor index has been defined as \[ESO(G) = \sum_{xy \in E(G)} (d_x + d_y) \sqrt{d_x^2 + d_y^2}\,.\] A molecular tree is a tree in which the maximum degree does not exceed \(4\). In this paper, we establish sharp upper and lower bounds for the \(ESO\) index in the class of molecular trees with order \(n\) and exactly \(k\) vertices of maximum degree \(\Delta \geq 2\). Moreover, we completely characterize the extremal trees attaining these bounds. Our findings contribute to the structural analysis of molecular trees and further the understanding of the elliptic Sombor index in chemical graph theory.
Zaslavsky (1991) characterized all single-element coextensions of graphic matroids in terms of a graphical structure called a biased graph. In this paper we characterize all orientations of a single-element coextension of a graphic matroid in terms of graphically defined orientations of its associated biased graph.
The exact deg-centric graph of a simple graph \(G\), denoted by \(G_{ed}\), is a graph constructed from \(G\) such that \(V(G_{ed}) = V(G)\) and \(E(G_{ed}) = \{v_iv_j: d_G(v_i,v_j) = deg_G(v_i)\}\). This research note presents the domination numbers of both the Jaco graph \(J_n(x)\) and the exact deg-centric graph of the family of Jaco graphs. The respective complement graphs are also addressed.
The first Zagreb index of a graph is one of the most important topological indices in chemical graph theory. It is also an important invariant of general graphs. The first Zagreb index of a graph is defined as the sum of the squares of the degrees of the vertices in the graph. The research on the Hamiltonian properties of a graph is an important topic in graph theory. Use the Diaz-Metcalf inequality, we in this paper present new sufficient conditions based on the first Zagreb index for the Hamiltonian and traceable graphs. In addition, using the ideas of obtaining the sufficient conditions, we also present an upper bound for the first Zagreb index of a graph. The graphs achieving the upper bound are also characterized.
This paper introduces the concept of the extended \(H\)-cover of a graph \(G\), denoted as \(G^*_H\) , as a generalization inspired by the extended double cover graphs discussed in Chen [1]. We explore the spectral properties and energy characteristics of \(G^*_H\), deriving formulae for the number of spanning trees in cases where both \(G\) and \(H\) are regular. Our investigation identifies several infinite families of equienergetic graphs and highlights instances of cospectral graphs within \(G^*_H\) . Additionally, we analyze various graph parameters related to the Indu-Bala product of graphs and the partial complement of the subdivision graph (PCSD) of \(G\).
A dominator coloring of a graph \(\mathscr{G}\) is a proper coloring where each vertex of \(\mathscr{G}\) is within the closed neighborhood of at least one vertex from each color class. The minimum number of color classes required for a dominator coloring of \(\mathscr{G}\) is termed the dominator chromatic number. Additionally, a total dominator coloring of a graph \(\mathscr{G}\) is a proper coloring in which every vertex dominates at least one color class other than its own. The minimum number of color classes needed for a total dominator coloring of \(\mathscr{G}\) is known as the total dominator chromatic number. In this paper, our objective is to derive findings concerning dominator and total dominator coloring of the duplication corresponding corona of specific graphs.
The topological index is a molecular property that is determined from a chemical compound’s molecular graph. Topological indices are numerical graph parameters that inform us about the topology of the graph and are generally graph invariants. In this paper, we consider some topological indices based on the second distance of each vertex of the graph \(\alpha\) and the number of unordered pairs of vertices \(\{s,q\} \subseteq V(\alpha)\) which are at distance \(3\) in \(\alpha\). These indices are called the leap Zagreb index and the Wiener polarity index, respectively. we compute these indices of \(R\)-vertex join and \(R\)-edge join of graphs.
