Work

The Sombor Index of Connected Graphs with Given Repetition Number and Cyclomatic Number

Year: 2026

Type: article

Abstract: The Sombor index of a graph G is SO(G)=∑uv∈E(G)(dG(u)2+dG(v)2)1/2, where dG(v) denotes the degree of v. The repetition number of G is the maximum multiplicity of an entry in its degree sequence. In this paper, we determine the maximum Sombor index among connected graphs with given repetition number r and cyclomatic number c, where c≥1 and r≥max{3,2c}. We characterize all graphs attaining the maximum value of the Sombor index. These extremal graphs consist of c cycles sharing one common vertex, together with r pendant vertices adjacent to the common vertex. For graphs outside the extremal family, we establish an explicit upper bound smaller than the maximum value. When r≥2c+1, we characterize all graphs attaining this bound, which also yields the second largest Sombor index. If r=2c, the bound is strict. We further prove a general lower bound in terms of the repetition number and a stronger bound involving both r and c for graphs with positive cyclomatic number. For c=1, the lower and upper bounds are both sharp for unicyclic graphs with given repetition number.

Source: Axioms

DOI: 10.3390/axioms15100757

Author: Hongbo Hua

Institution: Huai'an University

Language: English

Topic: Graph theory and applications

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $2,896

Funder: National Natural Science Foundation of China