On Clique Incidence Matrices and Derivatives of Clique Polynomials

TL;DR

This paper introduces clique incidence matrices as a generalization of classical incidence matrices, deriving identities and formulas for derivatives of clique polynomials, and discusses potential extensions and open questions.

Contribution

It presents a novel clique incidence matrix framework and derives new combinatorial formulas for derivatives of clique polynomials.

Findings

Derived two clique-counting identities

Obtained formulas for first and second derivatives of clique polynomials

Proposed open questions on higher derivatives

Abstract

The ordinary generating function of the number of complete subgraphs (cliques) of $G$, denoted by $C(G,x)$, is called the The clique polynomial of the graph $G$. In this paper, we first introduce some \emph{clique} incidence matrices associated by a simple graph $G$ as a generalization of the classical vertex-edge incidence matrix of $G$. Then, using these clique incidence matrices, we obtain two clique-counting identities that can be used for deriving two combinatorial formulas for the first and the second derivatives of clique polynomials. Finally, we conclude the paper with several open questions and conjectures about possible extensions of our main results for higher derivatives of clique polynomials.