$PC$-polynomial of graph

TL;DR

This paper introduces the $PC$-polynomial of graphs, explores its roots' properties, and relates them to growth rates of algebraic structures, providing bounds and asymptotic behaviors for these roots.

Contribution

It defines the $PC$-polynomial for graphs, analyzes its roots, and connects these roots to algebraic growth rates, including bounds and asymptotic properties.

Findings

Largest root $eta(G)$ relates to growth rate of partial commutative monoid.

Almost all graphs have roots near those of random graph $PC$-polynomial.

Derived bounds for maximum and minimum $eta(G)$ values.

Abstract

We define $PC$-polynomial of graph which is related to clique, (in)dependence and matching polynomials. The growth rate of partially commutative monoid is equal to the largest root $\beta(G)$ of $PC$-polynomial of the corresponding graph. The random algebra is defined in such way that its growth rate equals the largest root of $PC$-polynomial of random graph. We prove that for almost all graphs all sufficiently large real roots of $PC$-polynomial lie in neighbourhoods of roots of $PC$-polynomial of random graph. We show how to calculate the series expansions of the latter roots. The average value of $\beta(G)$ over all graphs with the same number of vertices is computed. We found the graphs on which the maximal value of $\beta(G)$ with fixed numbers of vertices and edges is reached. From this, we derive the upper bound of $\beta(G)$. Modulo one assumption, we do the same for minimal value of $\beta(G)$. We study the Nordhaus---Gaddum bounds of $\beta(G)+\beta(\bar{G})$ and $\beta(G)\beta(\bar{G})$.