On the maximum number of connected induced subgraphs of a graph

TL;DR

This paper characterizes the structure of graphs that maximize the number of connected induced subgraphs within various classes, considering parameters like degree, independence, and connectivity, filling gaps in existing literature.

Contribution

It provides a comprehensive characterization of extremal graphs for multiple graph classes based on various parameters, advancing theoretical understanding.

Findings

Identifies extremal graph structures for maximum connected induced subgraphs.

Provides characterizations for graphs with constraints like minimum degree and connectivity.

Fills gaps in the literature regarding extremal graph configurations.

Abstract

We characterise the structure of those graphs of a given order which maximise the number of connected induced subgraphs for seven different graph classes, each with other prescribed parameters like minimum degree, independence number, vertex cover number, vertex connectivity, edge connectivity, chromatic number, number of bridges, thereby contributing to filling a gap in the literature.