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Peer-reviewedCC BY 4.0Axioms

Average Fermat Difference on Unicyclic Graphs

  • Cheng Zeng1
  • Dongxu Wang1
  1. 1School of Mathematics and Information Science, Shandong Technology and Business University, Yantai 264003, China
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Abstract

For a connected graph, the Fermat difference of three vertices is defined as the minimum sum of their distances to a common vertex minus half the sum of their pairwise distances. In this paper, we investigate the average Fermat difference of unicyclic graphs with prescribed order and cycle length. By establishing a projection identity, we reduce the problem to a weighted cubic form on the unique cycle. We obtain an exact formula for cycles and establish a sharp lower bound together with a characterization of all extremal graphs attaining equality. We further derive a general upper bound for unicyclic graphs with prescribed order and cycle length. Finally, we determine the maximum average Fermat difference among all unicyclic graphs of fixed order and characterize the corresponding extremal graphs.

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Cite

BibTeXRIS
BibTeX
@article{zeng2026average,
  title = {{Average Fermat Difference on Unicyclic Graphs}},
  author = {Cheng Zeng and Dongxu Wang},
  journal = {Axioms},
  year = {2026},
  month = nov,
  doi = {10.3390/axioms15110824},
  url = {https://www.mdpi.com/2075-1680/15/11/824},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Average Fermat Difference on Unicyclic Graphs
AU  - Cheng Zeng
AU  - Dongxu Wang
PY  - 2026
DA  - 2026/11/05
JO  - Axioms
DO  - 10.3390/axioms15110824
UR  - https://www.mdpi.com/2075-1680/15/11/824
AB  - For a connected graph, the Fermat difference of three vertices is defined as the minimum sum of their distances to a common vertex minus half the sum of their pairwise distances. In this paper, we investigate the average Fermat difference of unicyclic graphs with prescribed order and cycle length. By establishing a projection identity, we reduce the problem to a weighted cubic form on the unique cycle. We obtain an exact formula for cycles and establish a sharp lower bound together with a characterization of all extremal graphs attaining equality. We further derive a general upper bound for unicyclic graphs with prescribed order and cycle length. Finally, we determine the maximum average Fermat difference among all unicyclic graphs of fixed order and characterize the corresponding extremal graphs.
SN  - 2075-1680
ER  -