The maximum spectral radius of $θ_{2,2,3}$-free graphs with given size
Jing Gao, Xueliang Li
TL;DR
This work resolves the maximum spectral radius problem for $\theta_{2,2,3}$-free graphs with a fixed number of edges by establishing a sharp upper bound $\lambda(G) \le \frac{1+\sqrt{4m-3}}{2}$ for $m\ge57$ and identifying the unique extremal graph achieving equality as $G \cong K_2 \vee \frac{m-1}{2}K_1$. Leveraging Perron-Frobenius theory, a careful vertex partition around a Perron vertex, and a sequence of structural lemmas, the authors rule out undesirable subgraph configurations and demonstrate that the extremal graph must be the join of $K_2$ with an independent set of size $\frac{m-1}{2}$. The result extends the spectral extremal theory for theta-free graphs and provides a complete characterization for the $\theta_{2,2,3}$-free case, contributing to a deeper understanding of how forbidding theta subgraphs constrains spectral radius and extremal structures.
Abstract
A theta graph $θ_{r,p,q}$ is the graph obtained by connecting two distinct vertices with three internally disjoint paths of length $r,p,q$, where $q\geq p\geq r\geq1$ and $p\geq2$. A graph is $θ_{r,p,q}$-free if it does not contain $θ_{r,p,q}$ as a subgraph. The maximum spectral radius of $θ_{1,p,q}$-free graphs with given size has been determined for any $q\geq p\geq2$. Zhai, Lin and Shu [Spectral extrema of graphs with fixed size: cycles and complete bipartite graphs, European J. Combin. 95 (2021) 103322] characterized the extremal graph with the maximum spectral radius of $θ_{2,2,2}$-free graphs having $m$ edges. In this paper, we consider the maximum spectral radius of $θ_{2,2,3}$-free graphs with size $m$ and characterize the extremal graph.