A Bivariate $B$-Restricted Clique Polynomial: From Local Neighborhoods to Global Expansion

TL;DR

This paper introduces a new bivariate clique polynomial that encodes local and global graph properties related to a subset B, providing bounds, stability results, and spectral insights with applications to graph expansion and homomorphisms.

Contribution

It defines the bivariate B-restricted clique polynomial, develops recurrence relations, proves stability for certain classes, and links it to spectral graph theory and homomorphism obstructions.

Findings

Monotonicity of the largest negative root under subgraph operations

Bounds on B-independence numbers, B-girth, and clique densities

Real-stability of the polynomial for specific graph classes

Abstract

Let $G$ be a finite simple graph and $B \subseteq V(G)$. We introduce the \emph{bivariate $B$-restricted clique polynomial} \[ C_B(G;x,y) = \sum_{\substack{K \subseteq V \\ K \text{ is a clique}}} x^{|K|} y^{|K \cap B|}, \] where the coefficient of $x^i y^j$ counts cliques of size $i$ with exactly $j$ vertices in $B$. This polynomial simultaneously captures combinatorial structure, local extremal properties, and spectral constraints associated with the subset $B$. \\ First, we develop vertex and edge deletion recurrences, generalizing classical clique polynomial results. These recurrences imply monotonicity for the largest negative root $\zeta_G(B;y)$ (viewed as a polynomial in $x$ for fixed $y \in [0,1]$) under induced and spanning subgraphs. From this, we derive bounds on $B$-independence numbers, $B$-girth, and clique densities restricted to $B$. \\ Next, we prove that for any integer $r \ge 1$, any $r$-connected $K_{r+3}$-free chordal graph $G$, and any subset $B \subseteq V(G)$, the bivariate clique polynomial $C_B(G;x,y)$ is real-stable. \\ Then, we connect $C_B(G;x,y)$ with spectral graph theory. For $(n,d,\lambda)$-graphs, expansion constraints via Tanner's inequality limit clique growth within $B$, yielding explicit bounds on coefficients and $\zeta_G(B;y)$. \\ Finally, we analyze weighted vertices and homomorphism obstructions in this framework, giving a general no-homomorphism criterion. We also conclude the paper with a couple of interesting open problems for young and motivated researchers.