Cut vertex and unicyclic graphs with the maximum number of connected induced subgraphs

TL;DR

This paper characterizes unicyclic graphs with a given number of nodes and cut vertices that maximize the number of connected induced subgraphs, revealing new structural insights and counterexamples to known correlations.

Contribution

It provides a complete description of maximal unicyclic graphs with respect to connected induced subgraphs for all valid node and cut vertex counts, including cases with girth constraints.

Findings

Identifies two types of maximal unicyclic graphs for given parameters.

Shows the failure of the negative correlation between connected subgraphs and Wiener index in certain unicyclic graphs.

Provides a characterization applicable to graphs with girth at most 4.

Abstract

Cut vertices are often used as a measure of nodes' importance within a network. They are those nodes whose failure disconnects a graph. Let N(G) be the number of connected induced subgraphs of a graph $G$. In this work, we investigate the maximum of N(G) where $G$ is a unicyclic graph with $n$ nodes of which $c$ are cut vertices. For all valid $n,c$, we give a full description of those maximal (that maximise N(.)) unicyclic graphs. It is found that there are generally two maximal unicyclic graphs. For infinitely many values of $n,c$, however, there is a unique maximal unicyclic graph with $n$ nodes and $c$ cut vertices. In particular, the well-known negative correlation between the number of connected induced subgraphs of trees and the Wiener index (sum of distances) fails for unicyclic graphs with $n$ nodes and $c$ cut vertices: for instance, the maximal unicyclic graph with $n=3,4\mod 5$ nodes and $c=n-5>3$ cut vertices is different from the unique graph that was shown by Tan et al.~[{\em The Wiener index of unicyclic graphs given number of pendant vertices or cut vertices}. J. Appl. Math. Comput., 55:1--24, 2017] to minimise the Wiener index. Our main characterisation of maximal unicyclic graphs with respect to the number of connected induced subgraphs also applies to unicyclic graphs with $n$ nodes, $c$ cut vertices and girth at most $g>3$, since it is shown that the girth of every maximal graph with $n$ nodes and $c$ cut vertices cannot exceed $4$.