Abstract:
This paper explores the extremal properties and bounds of two significant topological indices in graph theory: the Albertson and Sigma indices, with an emphasis on trees and bipartite graphs. We identify the unique trees that maximize and minimize the Albertson index, including stars and paths, and extend this characterization to bipartite graphs. In this paper, we investigate the sharp upper and lower bounds of topological indices for a given degree sequence $\mathscr{D}=(d_1,d_2,\ldots,d_n)$. We derive exact lower and upper bounds for the Albertson index and Sigma index based on a non-increasing degree sequence $\mathscr{D}=(d_1,d_2,\ldots,d_n)$. Establishing such bounds is a fundamental challenge in the study of topological indices, as these results reveal inherent relationships among various indices. For generating bipartite graphs and tournaments with prescribed degree sequences, analyzing their mixing times and convergence properties. The sharp upper and lower bounds for the Sigma index based on degree sequences, providing a deeper understanding of its behavior in trees. Our findings offer novel insights into graph irregularity measures, supported by rigorous proofs and computational algorithms for evaluating these indices in random trees and forests. These results contribute to the understanding of extremal properties and combinatorial structures in graph theory, with applications in chemical graph theory and network analysis.
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