Extremal Results on ℓ -Connected Graphs or Pancyclic Graphs Based on Wiener-Type Indices

Jing Zeng, Hechao Liu and Lihua You ()
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Jing Zeng: School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
Hechao Liu: School of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, China
Lihua You: School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Mathematics, 2024, vol. 13, issue 1, 1-16

Abstract: A graph of order n is called pancyclic if it contains a cycle of length y for every 3 ≤ y ≤ n . The connectivity of an incomplete graph G , denoted by κ ( G ) , is min { | W | | W i s a v e r t e x c u t o f G } . A graph G is said to be ℓ -connected if the connectivity κ ( G ) ≥ ℓ . The Wiener-type indices of a connected graph G are W g ( G ) = ∑ { s , t } ⊆ V ( G ) g ( d G ( s , t ) ) , where g ( x ) is a function and d G ( s , t ) is the distance in G between s and t . In this note, we first determine the minimum and maximum values of W g ( G ) for ℓ -connected graphs. Then, we use the Wiener-type indices of graph G , the Wiener-type indices of complement graph G ¯ with minimum degree δ ( G ) ≥ 2 or δ ( G ) ≥ 3 to give some sufficient conditions for connected graphs to be pancyclic. Our results generalize some existing results of several papers.

Keywords: ?-connected graph; pancyclic graph; Wiener-type index; sufficient condition (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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