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

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 ρₜ(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.