Maximizing subgraph counts in regular graphs

TL;DR

The paper addresses the problem of maximizing the $H$-density in $d$-regular graphs by recasting subgraph counts as injective and ordinary homomorphism densities and then embedding the problem into a spectral optimization framework. It introduces degree-$|V(H)|$ polynomials $p_H( ext{λ},d)$ that bound injective counts in terms of the spectrum $ ext{σ}(G)$ and proves that, for large $d$, extremizers are disjoint unions of either $K_{d,d}$ (when $H$ is bipartite) or $K_{d+1}$ (when $H$ is non-bipartite), with sharpness and uniqueness results tied to the spectral characterizations of these graphs. In the special case $H=C_5$, the threshold is $d eq 7$ and the unique optimal $3$-regular graph is the Petersen graph, while for larger $d$ the optimizers align with the corresponding clique structures; the paper also develops a general methodology and discusses near-optimal behavior and potential extensions to constrained extremal problems. Overall, the work provides a unified toolbox that blends homomorphism inequalities with spectral optimization to tackle Turán-type questions in regular graphs and offers a pathway to analyze a broad class of subgraph-count extremal problems.

Abstract

Given a graph $H$, we investigate the $d$-regular graphs $G$ with the highest $H$-density. We reframe the problem as a continuous optimization problem on the eigenvalues of $G$ by relating injective homomorphism numbers from $H$ and homomorphism numbers from quotient graphs of $H$. For almost all $H$, this relation has non-spectral terms, which require bounding by spectral terms in a way that is sharp at the optimal graph. For bipartite $H$ and $d$ large enough, we show $G$ consists of disjoint copies of $K_{d,d}$. For non-bipartite $H$ and $d$ sufficiently large, $G$ is a collection of disjoint $K_{d+1}$ graphs. For $H=C_5$ and $d=3$, disjoint Petersen graphs emerge.

Figures (3)

• Figure 1: A 3-regular graph with 5 edges through $e$ and a cut edge.
• Figure 2: 4-regular graphs with high $C_5$ density
• Figure 3: Maximal cases for $x_1=z_1,~x_2=z_2$ and $x_1=z_1,~x_2 \neq z_2$.

Theorems & Definitions (26)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Proposition 1.5
• Corollary 1.6
• Proposition 1.7
• Definition 2.1
• Proposition 2.2: lovasz2012large
• Proposition 2.3: lovasz2012large