On the sum of eccentric subgraph numbers of the vertices of a connected graph

@misc{de2026eccentric,
  title = {{On the sum of eccentric subgraph numbers of the vertices of a connected graph}},
  author = {Akash De and Kamal Lochan Patra},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.19694},
  eprint = {2610.19694},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.19694},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - On the sum of eccentric subgraph numbers of the vertices of a connected graph
AU  - Akash De
AU  - Kamal Lochan Patra
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.19694
UR  - https://arxiv.org/abs/2610.19694
AB  - Let G be a connected graph. The eccentric subgraph number ε(v) of a vertex v of G is defined as ε(v)=min{f_G(v,u): u∈ V(G)}, where f_G(v,u) is the number of connected subgraphs of G containing both v and u. The eccentric subgraph index ES(G) of G is defined as ES(G)=∑_v∈ V(G)ε(v). In this paper, we obtain the unicyclic graph that maximizes the eccentric subgraph index among all unicyclic graphs on n vertices with a fixed girth. The graph that maximizes the eccentric subgraph index among all unicyclic graphs on n vertices is also obtained. We then characterize the trees that maximize the eccentric subgraph index over trees on n vertices with fixed diameter. Among all connected graphs on n vertices with a fixed number of pendant vertices, the graph which maximizes the eccentric subgraph index is also characterized.
ER  -