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

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.