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. These are nodes whose failure disconnects a connected 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 5\) nodes and \ (c=n-5>3\) cut vertices is different from the unique graph that was shown by Tan et al. 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\).