A $B$-Restricted Clique Polynomial and Connections to Tanner's Inequality

TL;DR

This paper introduces the $B$-restricted clique polynomial as a versatile tool for analyzing graph structure, spectral properties, and homomorphism constraints, unifying various graph invariants and inequalities.

Contribution

It develops a comprehensive deletion theory, establishes bounds on graph parameters, connects spectral graph theory with clique growth, and encodes homomorphism constraints within the $B$-clique polynomial framework.

Findings

Monotonicity of the largest negative root under subgraph relations.

Explicit bounds on independence, chromatic numbers, and girth using the root.

Spectral bounds on clique coefficients in expander graphs.

Abstract

Let $G$ be a finite simple graph and $B \subseteq V(G)$. We study the \emph{$B$-restricted clique polynomial} $C_B(G;x)$, including its weighted version allowing vertex multiplicities, as a versatile tool to capture structural properties of vertex subsets. First, we develop a complete deletion theory for $C_B(G;x)$, including vertex and edge recurrences that generalize classical clique polynomial results. These recurrences yield monotonicity principles for the largest negative root $\zeta_G(B)$: it is monotone under induced subgraphs and reverse-monotone under spanning subgraphs. Consequently, we derive explicit bounds on $B$-independence numbers, chromatic numbers, $B$-girth, and Hamiltonicity constraints, showing that $\zeta_G(B)$ serves as a unifying local invariant. Next, we connect $B$-clique polynomials to spectral graph theory. For $(n,d,\lambda)$-graphs, spectral techniques, including the Expander Mixing Lemma and Tanner's inequality, provide uniform bounds on $B$-restricted clique coefficients, demonstrating that clique growth within $B$ is naturally controlled by the spectral gap. Finally, we show that weighted $B$-clique polynomials encode \emph{homomorphism constraints}. Specifically, if $f: G \to H$ is a surjective homomorphism mapping $B_G$ onto $B_H$, then $\zeta_G(B_G) \ge \zeta_H(B_H)$, yielding a local \emph{no-homomorphism criterion} based on $B$-roots. Overall, $C_B(G;x)$ provides a unified framework capturing combinatorial, spectral, and homomorphic information in vertex-restricted analysis, highlighting its power for both global and local structural insights.