The high order spectral radius of graphs without long cycles or paths

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 ${\rho }_t(G)$. 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.