Maximising the number of connected induced subgraphs of unicyclic graphs

Denote by G(n, c, g, k) the set of all connected graphs of order n, having c cycles, girth g, and k pendant vertices. In this paper, we give a partial characterisation of the structure of those graphs in G(n, c, g, k) maximising the number of connected induced subgraphs. For the special case where c = 1, we find a complete characterisation of all maximal unicyclic graphs. We also derive a precise formula for the corresponding maximum number given the following parameters: (1) order, girth, and number of pendant vertices; (2) order and girth; (3) order.