Extremal problems for the p-spectral radius of graphs

The p-spectral radius of a graph G of order n is defined for any real number p ≥ 1 as The most remarkable feature of λ(p) is that it seamlessly joins several other graph parameters, e.g., λ(1) is the Lagrangian, λ(2) is the spectral radius and λ(∞)/2 is the number of edges. This paper presents solutions to some extremal problems about λ(p), which are common generalizations of corresponding edge and spectral extremal problems. Let Tr (n) be the r-partite Turán graph of order n. Two of the main results in the paper are: (I) Let r ≥ 2 and p \u3e 1. If G is a Kr+1-free graph of order n, then λ(p) (G) \u3c λ(p) (Tr (n)), unless G=Tr(n). (II) Let r ≥ 2 and p \u3e1. If G is a graph of order n, with λ (p) (G) \u3e λ(p) (Tr (n)), then G has an edge contained in at least cnr-1 cliques of order r+1, where c is a positive number depending only on p and r