Average distance, minimum degree, and spanning trees

Peter Dankelmann; Roger Entringer

doi:10.1002/(SICI)1097-0118(200001)33:1<1::AID-JGT1>3.0.CO;2-L

Average distance, minimum degree, and spanning trees

Peter Dankelmann, Roger Entringer

Research output: Contribution to journal › Article › peer-review

69 Citations (Scopus)

Abstract

The average distance μ(G) of a connected graph G of order n is the average of the distances between all pairs of vertices of G, i.e., μ(G) = (ⁿ₂)^-1 ∑{x,y}⊂V(G) d_G(x,y), where V(G) denotes the vertex set of G and d_G(x, y) is the distance between x and y. We prove that every connected graph of order n and minimum degree δ has a spanning tree T with average distance at most n/δ+1 + 5. We give improved bounds for K₃-free graphs, C₄-free graphs, and for graphs of given girth.

Original language	English
Pages (from-to)	1-13
Number of pages	13
Journal	Journal of Graph Theory
Volume	33
Issue number	1
DOIs	https://doi.org/10.1002/(SICI)1097-0118(200001)33:1<1::AID-JGT1>3.0.CO;2-L
Publication status	Published - Jan 2000
Externally published	Yes

Keywords

Average distance
Graph
Mininum degree
Spanning tree

ASJC Scopus subject areas

Geometry and Topology

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10.1002/(SICI)1097-0118(200001)33:1<1::AID-JGT1>3.0.CO;2-L

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AB - The average distance μ(G) of a connected graph G of order n is the average of the distances between all pairs of vertices of G, i.e., μ(G) = (n2)-1 ∑{x,y}⊂V(G) dG(x,y), where V(G) denotes the vertex set of G and dG(x, y) is the distance between x and y. We prove that every connected graph of order n and minimum degree δ has a spanning tree T with average distance at most n/δ+1 + 5. We give improved bounds for K3-free graphs, C4-free graphs, and for graphs of given girth.

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KW - Graph

KW - Mininum degree

KW - Spanning tree

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Average distance, minimum degree, and spanning trees

Abstract

Keywords

ASJC Scopus subject areas

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