About AutoGraphiX Conjecture on Domination Number and Remoteness of Graphs

Lidan Pei ()
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Lidan Pei: School of Mathematics and Statistics, Hefei Normal University, Hefei 230601, China

Mathematics, 2022, vol. 10, issue 19, 1-12

Abstract: A set D ⊆ V ( G ) is called a dominating set if N [ v ] ∩ D ≠ ∅ for every vertex v in graph G . The domination number γ ( G ) is the minimum cardinality of a dominating set of G . The proximity π ( v ) of a vertex v is the average distance from it to all other vertices in graph. The remoteness ρ ( G ) of a connected graph G is the maximum proximity of all the vertices in graph G . AutoGraphiX Conjecture A.565 gives the sharp upper bound on the difference between the domination number and remoteness. In this paper, we characterize the explicit graphs that attain the upper bound in the above conjecture, and prove the improved AutoGraphiX conjecture.

Keywords: AutoGraphiX conjecture; domination number; remoteness (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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