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

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 $4$.