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Cut vertices are often used as a measure of nodes' importance within anetwork. They are those nodes whose failure disconnects a graph. Let N(G) bethe number of connected induced subgraphs of a graph $G$. In this work, weinvestigate the maximum of N(G) where $G$ is a unicyclic graph with $n$ nodesof which $c$ are cut vertices. For all valid $n,c$, we give a full descriptionof those maximal (that maximise N(.)) unicyclic graphs. It is found that thereare 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 betweenthe number of connected induced subgraphs of trees and the Wiener index (sum ofdistances) fails for unicyclic graphs with $n$ nodes and $c$ cut vertices: forinstance, 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 etal.~[{\em The Wiener index of unicyclic graphs given number of pendant verticesor cut vertices}. J. Appl. Math. Comput., 55:1--24, 2017] to minimise theWiener index. Our main characterisation of maximal unicyclic graphs withrespect to the number of connected induced subgraphs also applies to unicyclicgraphs with $n$ nodes, $c$ cut vertices and girth at most $g>3$, since it isshown that the girth of every maximal graph with $n$ nodes and $c$ cut verticescannot exceed $4$.

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