The Exponential Hyper-Zagreb Indices and Structural Properties of Trees and Bipartite Graphs

TL;DR

This paper studies the structural properties of trees and bipartite graphs using Hyper-Zagreb indices and combinatorial methods, revealing how high-degree vertices influence graph indices and characterizing extremal structures.

Contribution

It introduces new bounds and characterizations for Hyper-Zagreb indices in trees and bipartite graphs, extending to exponential forms and exploring extremal and structural properties.

Findings

Presence of high-degree vertices affects Hyper-Zagreb indices

Characterization of extremal trees via degree sequence majorization

Bounds on independence number and competition numbers in bipartite graphs

Abstract

In this paper, we investigate the structural properties of trees and bipartite graphs through the lens of topological indices and combinatorial graph theory. We focus on the First and Second Hyper-Zagreb indices, $HM_1(G)$ and $HM_2(G)$, for trees $T \in T(n, \Delta)$ with $n$ vertices and maximum degree $\Delta$. Key propositions demonstrate that the presence of end-support or support vertices of degree at least three, distinct from a vertex of maximum degree, implies the existence of another tree $T' \in T(n, \Delta)$ with strictly smaller Hyper-Zagreb indices. These results are extended to exponential forms, highlighting the influence of high-degree vertices. Additionally, we explore structural characterizations of trees via degree sequence majorization and $S$-order, establishing conditions for the first and last trees in specific classes. For bipartite graphs, we examine equitable coloring, cycle lengths, and $k$-redundant tree embeddings, supported by theorems on connectivity and minimum degree constraints. The paper also addresses the independence number of bipartite graphs, exterior covers, and competition numbers of complete $r$-partite graphs, providing bounds and structural insights. Finally, we discuss Markov-chain algorithms for generating bipartite graphs and tournaments with prescribed degree sequences, analyzing their mixing times and convergence properties. These results contribute to the understanding of extremal properties and combinatorial structures in graph theory, with applications in chemical graph theory and network analysis.