Abstract
A vertex v is a peripheral vertex in G if its eccentricity is equal to its diameter, and periphery P ( G ) is a subgraph of G induced by its peripheral vertices. Further, a vertex v in G is a central vertex if e ( v ) = r a d ( G ) , and the subgraph of G induced by its central vertices is called center C ( G ) of G . Average eccentricity is the sum of eccentricities of all of the vertices in a graph divided by the total number of vertices, i.e., a v e c ( G ) = { 1 n ∑ e G ( u ) ; u ∈ V ( G ) } . If every vertex in G is central vertex, then C ( G ) = G , and hence, G is self-centered. In this report, we find the center, periphery and average eccentricity for the convex polytopes.
Highlights
In the facility location problem, we select a site according to some standard judgment
Average eccentricity is the sum of eccentricities of all of the vertices in a graph divided by the total number of vertices, i.e., avec( G ) =
A vertex v is a peripheral vertex in G if its eccentricity is equal to its diameter, and periphery
Summary
In the facility location problem, we select a site according to some standard judgment. The theme of distance is used to check the symmetry of graphs It provides a base for many useful graph parameters, like radius, diameter, metric dimension, eccentricity, center and periphery, etc. The idea of self-centered graphs is presented and elaborated by Ando, Akiyama and Avis individually [7]. The extremal size of a connected self-centered graph with p vertices and r radius is explained by F. For a connected graph G, the eccentricity e(v) of a vertex v is its distance to a vertex farthest from v. The diameter diam( G ) of G is the maximum eccentricity among all vertices of G. We discuss the center, periphery and average eccentricity for families of convex polytope graphs, An , Sn and Tn
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