Enumerating simple paths from connected induced subgraphs

TL;DR

This paper introduces an exact formula for counting simple paths between two vertices in a graph, utilizing the adjacency matrix of connected induced subgraphs, applicable to weighted and directed graphs, and relates Hamiltonian paths to dominating connected sets.

Contribution

It provides a novel formula for enumerating simple paths that incorporates connected induced subgraphs and extends to weighted and directed graphs.

Findings

Derived an exact generating series formula for simple paths.

Linked Hamiltonian paths and cycles to dominating connected sets.

Applicable to weighted and directed graphs.

Abstract

We present an exact formula for the ordinary generating series of the simple paths between any two vertices of a graph. Our formula involves the adjacency matrix of the connected induced subgraphs and remains valid on weighted and directed graphs. As a particular case, we obtain a relation linking the Hamiltonian paths and cycles of a graph to its dominating connected sets.