Abstract
A graph is a block graph if its blocks are all cliques. In this paper, we study the average eccentricity of block graphs from the perspective of block order sequences. An equivalence relation is established under the block order sequence and used to prove the lower and upper bounds of the eccentricity on block graphs. The result is that the lower and upper bounds of the average eccentricity on block graphs are 1 and 1n⌊34n2−12n⌋, respectively, where n is the order of the block graph. Finally, we devise a linear time algorithm to calculate the block order sequence.
Highlights
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By Theorem 9, the transformation does not decrease the average eccentricity, so the upper bound on block graph set G(S) must be achieved by a path-like block graph in G(S)
Let PGmin max be path-like block graphs having, respectively, minimum and maximum average eccentricity on the set of block graphs with block order sequence S
Summary
Relying on a polynomial algorithm for the All-Pairs Shortest Paths Problem [2], the average eccentricity can be calculated in time complexity O(n3 log n) where n is the order of the graph. The properties, formulas, and bounds on the average eccentricity have recently been studied intensely [3–12]. Ahmad et al [18] studied the chemical graphs of copper oxide and carbon graphite They computed and gave close formulas of eccentricity based topological indices, such as total eccentricity index and average eccentricity index for chemical graphs of carbon graphite and copper oxide.
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