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Average Distance, Independence Number, and Spanning Trees
Author(s) -
Mukwembi Simon
Publication year - 2014
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.21758
Subject(s) - mathematics , combinatorics , independence number , corollary , spanning tree , conjecture , upper and lower bounds , independence (probability theory) , minimum degree spanning tree , discrete mathematics , minimum spanning tree , graph , statistics , mathematical analysis
Let G be a connected graph of order n and independence number α. We prove that G has a spanning tree with average distance at most2 3 α , if n ≤ 2 α − 1 , and at most α + 2 , if n > 2 α − 1 . As a corollary, we obtain, for n sufficiently large, an asymptotically sharp upper bound on the average distance of G in terms of its independence number. This bound, apart from confirming and improving on a conjecture of Graffiti [8], is a strengthening of a theorem of Chung [1], and that of Fajtlowicz and Waller [8], on average distance and independence number of a graph.