Digraph Polynomials for Counting Cycles and Paths

TL;DR

This paper introduces new polynomial invariants for directed graphs, called cycle and path polynomials, which count cycles and paths and satisfy recurrence relations, generalizing existing polynomials.

Contribution

It defines the cycle and path polynomials for directed graphs, establishes their recurrence relations, and introduces a most general digraph polynomial related to these concepts.

Findings

Cycle and path polynomials count cycles and paths in directed graphs.

Recurrence relations connect these polynomials to elementary graph operations.

A general digraph polynomial is defined and expressed explicitly.

Abstract

Many polynomial invariants are defined on graphs for encoding the combinatorial information and researching them algebraically. In this paper, we introduce the cycle polynomial and the path polynomial of directed graphs for counting cycles and paths, respectively. They satisfy recurrence relations with respect to elementary edge or vertex operations. They are related to other polynomials and can also be generalized to the bivariate cycle polynomial, the bivariate path polynomial and the trivariate cycle-path polynomial. And a most general digraph polynomial satisfying such a linear recurrence relation is recursively defined and shown to be co-reducible to the trivariate cycle-path polynomial. We also give an explicit expression of this polynomial.