Graphs having extremal monotonic topological indices with bounded vertex $k$-partiteness

TL;DR

This paper characterizes extremal graphs with bounded vertex k-partiteness that optimize various topological indices, providing insights into their structure and extremal properties.

Contribution

It introduces the concepts of monotonic decreasing and increasing topological indices and characterizes extremal graphs for several indices with fixed vertex k-partiteness.

Findings

Identified extremal graphs for Wiener index, Harry index, and reciprocal degree distance.

Determined graphs with maximum connective eccentricity and Zagreb indices.

Provided structural characterizations for graphs with bounded vertex k-partiteness.

Abstract

The vertex $k$-partiteness $v_k(G)$ of graph $G$ is defined as the fewest number of vertices whose deletion from $G$ yields a $k$-partite graph. In this paper, we introduce two concepts: monotonic decreasing topological index and monotonic increasing topological index, and characterize the extremal graphs having the minimum Wiener index, the maximum Harry index, the maximum reciprocal degree distance, the minimum eccentricity distance sum, the minimum adjacent eccentric distance sum index, the maximum connective eccentricity index, the maximum Zagreb indices among graphs with a fixed number $n$ of vertices and fixed vertex $k$-partiteness, respectively.