A note on the second-largest number of dissociation sets in connected graphs

TL;DR

This paper investigates the second-largest number of dissociation sets in connected graphs, providing a positive answer to a recent open question about extremal graph structures.

Contribution

It identifies the second-largest number of dissociation sets in connected graphs and characterizes the extremal graphs achieving this bound.

Findings

Confirmed the second-largest number of dissociation sets in connected graphs.

Characterized the extremal graphs for the second-largest case.

Abstract

A subset of vertices is called a dissociation set if it induces a subgraph with vertex degree at most one. Recently, Yuan et al. established the upper bound of the maximum number of dissociation sets among all connected graphs of order n and characterized the corresponding extremal graphs.They also proposed a question regarding the second-largest number of dissociation sets among all connected graphs of order n and the corresponding extremal graphs. In this paper, we give a positive answer to this question.