The Sombor Index of Connected Graphs with Given Repetition Number and Cyclomatic Number
- 1Faculty of Mathematics and Physics, Huai’an University, Huai’an 223003, China
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.
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BibTeX
@article{hua2026sombor,
title = {{The Sombor Index of Connected Graphs with Given Repetition Number and Cyclomatic Number}},
author = {Hongbo Hua},
journal = {Axioms},
year = {2026},
month = oct,
doi = {10.3390/axioms15100757},
url = {https://www.mdpi.com/2075-1680/15/10/757},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - The Sombor Index of Connected Graphs with Given Repetition Number and Cyclomatic Number
AU - Hongbo Hua
PY - 2026
DA - 2026/10/10
JO - Axioms
DO - 10.3390/axioms15100757
UR - https://www.mdpi.com/2075-1680/15/10/757
AB - 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.
SN - 2075-1680
ER -