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Mathematics > Combinatorics

arXiv:1812.02422 (math)
[Submitted on 6 Dec 2018 (v1), last revised 17 Sep 2019 (this version, v2)]

Title:Graphs and unicyclic graphs with extremal number of connected induced subgraphs

Authors:Audace A. V. Dossou-Olory
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Abstract:Over all graphs (or unicyclic graphs) of a given order, we characterise those graphs that minimise or maximise the number of connected induced subgraphs. For each of these classes, we find that the graphs that minimise the number of connected induced subgraphs coincide with those that are known to maximise the Wiener index (the sum of the distances between all unordered pairs of vertices), and vice versa. For every $k$, we also determine the connected graphs that are extremal with respect to the number of $k$-vertex connected induced subgraphs. We show that, in contrast to the minimum which is uniquely realised by the path, the maximum value is attained by a rich class of connected graphs.
Comments: 17 pages, 4 figures
Subjects: Combinatorics (math.CO)
MSC classes: Primary 05C30, secondary 05C35, 05C05
Cite as: arXiv:1812.02422 [math.CO]
  (or arXiv:1812.02422v2 [math.CO] for this version)
  https://doi.org/10.48550/arXiv.1812.02422

Submission history

From: Audace Amen Vioutou Dossou-Olory [view email]
[v1] Thu, 6 Dec 2018 09:36:49 UTC (17 KB)
[v2] Tue, 17 Sep 2019 07:23:46 UTC (17 KB)
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