On Graphs with Maximal Sum of α-th Degree Powers

Let G be a simple graph. The sum of the α-th degree powers of G, denoted by Mα(G), is obtained by summing the α-th powers of all vertex degrees. We use Gn,m to denote the set of all graphs having n vertices and m edges. Extremal problems concerning M2(G) have a long history in combinatorics and graph theory. In 1971, Katz (Israel J. Math., 1971) first initiated the study of maximizing M2(G) within Gn,m. After nearly 40 years, Ábrego et al. (J. Inequal. Pure and Appl. Math., 2009) completely resolved this problem. In this paper, we extend the power exponent from α=2 to α>2, investigating the problem of determining the graph that maximizes the value of Mα(G) within Gn,m. We show that the graph that maximizes Mα(G) when α>2 and 1≤m≤14n2 is the quasi-star graph, which belongs to a special class of threshold graphs (defined in the Introduction).