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On the number of connected subgraphs of graphs

Abstract

For a connected graph G,  we denote the number of connected subgraphs of G by F(G). For a tree T, F(T) has been studied extensively and it has been observed that F(T) has a reverse correlation with Wiener index of T. We call F(G),  the subgraph index of G. In this paper, we study the subgraph index of unicyclic graphs and graphs with fixed number of pendant vertices. We obtain the unicyclic graphs which extremize the subgraph index over all unicyclic graphs on n vertices. The graphs which extremize the subgraph index among all unicyclic graphs with fixed girth are also obtained. Among all connected graphs on n vertices with fixed number of pendant vertices, the graph which minimizes and the graph which maximizes the subgraph index are characterized.

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Correspondence to Kamal Lochan Patra.

Additional information

Communicated by Sharad S Sane, PhD.

D. Pandey:Supported by UGC Fellowship scheme (Sr. No. 2061641145), Government of India.

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Pandey, D., Patra, K.L. On the number of connected subgraphs of graphs. Indian J Pure Appl Math 52, 571–583 (2021). https://doi.org/10.1007/s13226-021-00061-4

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  • DOI: https://doi.org/10.1007/s13226-021-00061-4

Keywords

  • Tree
  • Unicyclic graph
  • Girth
  • Subtree core
  • Wiener index

Mathematics Subject Classification

  • 05C05
  • 05C07
  • 05C30
  • 05C35