The high order spectral radius of graphs without long cycles or paths
Yuntian Wang, Lizhu Sun, Changjiang Bu
TL;DR
The paper develops a high order spectral framework for graphs by introducing the $t$-clique spectral radius $\rho_t(G)$, defined via the $t$-clique tensor $A_t(G)$, and extends Erdős-Gallai-type extremal results to this setting. Using a Perron-vector analysis and rotation operations $G_{u\to v}$, the authors show that extremal graphs for forbidding long cycles ($C_{\ge k}$-free) or long paths ($P_k$-free) must have a join-structure, specifically $S_{n,(k-1)/2}$ when $k$ is odd and $S_{n,(k-2)/2}^{+}$ when $k$ is even, for sufficiently large $n$. They also establish uniqueness within the relevant extremal families for a broad range of $t$ (namely $2\le t \le \lfloor (k+1)/2\rfloor$ in the cycle case) and show that the same conclusions hold in the path-free setting, with proofs mirroring each other. Overall, the work advances spectral extremal graph theory by providing high order (clique) analogues of Erdős-Gallai theorems and characterizing the extremal structures that maximize $\rho_t(G)$ under long-cycle and long-path forbiddance.
Abstract
In 1959, Erdős and Gallai established two classic theorems, which determine the maximum number of edges in an $n$-vertex graph with no cycles of length at least $k$, and in an $n$-vertex graph with no paths on $k$ vertices, respectively. Subsequently, generalized and spectral versions of the Erdős-Gallai theorems have been investigated. A concept of a high order spectral radius for graphs was introduced in 2023, defined as the spectral radius of a tensor and termed the $t$-clique spectral radius $ρ_t(G)$. In this paper, we establish a high order spectral version of Erdős-Gallai theorems by employing the $t$-clique spectral radius, i.e., we determine the extremal graphs that attain the maximum $t$-clique spectral radius in the $n$-vertex graphs with no cycles of length at least $k$ and in the $n$-vertex graphs with no paths on $k$ vertices, respectively.