Work

On Graphs with Maximal Sum of α-th Degree Powers

Year: 2026

Type: article

Abstract: 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).

Source: Mathematics

DOI: 10.3390/math14101613

Authors: Bo-Jun Yuan, Jia-Jin Wang

Institution: Zhejiang University of Science and Technology

Language: English

Topic: Graph theory and applications

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,137

Funder: National Natural Science Foundation of China