Average eccentricity, minimum degree and maximum degree in graphs

P. Dankelmann () and F. J. Osaye
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P. Dankelmann: University of Johannesburg
F. J. Osaye: University of Johannesburg

Journal of Combinatorial Optimization, No 0, 16 pages

Abstract: Abstract Let G be a connected finite graph with vertex set V(G). The eccentricity e(v) of a vertex v is the distance from v to a vertex farthest from v. The average eccentricity of G is defined as $$\frac{1}{|V(G)|}\sum _{v \in V(G)}e(v)$$ 1 | V ( G ) | ∑ v ∈ V ( G ) e ( v ) . We show that the average eccentricity of a connected graph of order n, minimum degree $$\delta $$ δ and maximum degree $$\Delta $$ Δ does not exceed $$\frac{9}{4} \frac{n-\Delta -1}{\delta +1} \big ( 1 + \frac{\Delta -\delta }{3n} \big ) + 7$$ 9 4 n - Δ - 1 δ + 1 ( 1 + Δ - δ 3 n ) + 7 , and this bound is sharp apart from an additive constant. We give improved bounds for triangle-free graphs and for graphs not containing 4-cycles.

Keywords: Average eccentricity; Distance; Minimum degree; Maximum degree; Eccentricity; Graph (search for similar items in EconPapers)
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DOI: 10.1007/s10878-020-00616-x

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