Ordinary and spectral extremal problems on vertex disjoint copies of even fans
Yiting Cai, Bo Zhou
TL;DR
The paper addresses the extremal problem for forbidding $t$ vertex-disjoint copies of the even fan $P_1\vee P_{2k}$ by determining both the maximum edge number and the maximum spectral radius among $F$-free graphs of order $n$, for fixed $t\ge1$ and $k\ge3$. It derives an exact-size bound $e(G)\le f(n,t)$ and characterizes the extremal graphs as $K_{t-1}\vee H$ with $H$ drawn from the family $\mathcal{K}_{n_1,n-n_1-t+1}^{k-1}(P_{2k})$, with $n_1$ selected by precise modular conditions; a parallel spectral result shows the same join-structure governs the extremal graphs for the spectral radius, analyzed via equitable partitions and quotient matrices. The authors employ progressive induction (Simonovits framework), stability tools, and Rayleigh-quotient arguments to prove both the size and spectral classifications and to pinpoint when the two extremal notions coincide. These results extend the $t=1$ case and connect with Turán-type stability phenomena, offering a detailed structural description of extremal graphs in terms of a near-bipartite join augmented by a $P_{2k}$-free, regular (or nearly regular) component.
Abstract
Let $\mathrm{ex}(n, F)$ and $\mathrm{spex}(n, F)$ be the maximum size and spectral radius among all $F$-free graphs with fixed order $n$, respectively. A fan is a graph $P_1\vee P_{s}$ (join of a vertex and a path of order $s$) for $s\ge 3$, and it is called an even fan if $s$ is even. In this paper, we study $\mathrm{ex}(n,t(P_1\vee P_{2k}))$, $\mathrm{spex}(n,t(P_1\vee P_{2k}))$ with $t\ge 1$ and $k\ge 3$ and characterize the corresponding extremal graphs for sufficiently large $n$.