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 λ (G) = max 2 ∑ {i,j}∈E(G) xixj : x1, . . . , xn ∈ R and |x1| + · · ·+ |xn| = 1  . 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 > 1. If G is a Kr+1-free graph of order n, then λ (G) < λ (Tr (n)) , unless G = Tr (n) . (II) Let r > 2 and p > 1. If G is a graph of order n, with λ (G) > λ (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.