The Second Zagreb Indices of Graphs with Given Degree Sequences

TL;DR

This paper studies the maximum second Zagreb index in graphs with specified degree sequences, especially bicyclic sequences, and explores relationships among extremal graphs with different degree sequences.

Contribution

It introduces new results on extremal graphs with maximum second Zagreb indices for given degree sequences, including bicyclic sequences, and analyzes their interrelations.

Findings

Identified extremal graphs with maximum second Zagreb indices for specific degree sequences.

Established relations among second Zagreb indices of extremal graphs with different degree sequences.

Provided characterizations of extremal graphs in the context of bicyclic sequences.

Abstract

The second Zagreb index of a graph G is denoted by $M_2(G)=\sum_{uv\in E(G)}d(u)d(v)$. In this paper, we investigate properties of the extremal graphs with the maximum second Zagreb indices with given graphic sequences, in particular graphic bicyclic sequences. Moreover, we obtain the relations of the second Zagreb indices among the extremal graphs with different degree sequences.