Vertex-to-clique detour distance in graphs | [Journal of Prime Research in Mathematics • 2016]

Author(s):

1. I. KEERTHI ASIR: Department of Mathematics, St. Xavier's College (Autonomous), Palayamkottai 627002, Tamil Nadu, India.

2. S. ATHISAYANATHAN: Department of Mathematics, St. Xavier's College(Autonomous),Palayamkottai 627002, Tamil Nadu, India

Abstract:

Let v be a vertex and C a clique in a connected graph G. A vertex-to-clique u ¡ C path P is a u ¡ v path, where v is a vertex in C such that P contains no vertices of C other than v. The vertex-to-clique distance, d(u; C) is the length of a smallest u¡C path in G. A u¡C path of length d(u; C) is called a u ¡ C geodesic. The vertex-to-clique eccentricity e1(u) of a vertex u in G is the maximum vertex-to-clique distance from u to a clique C 2 ³, where ³ is the set of all cliques in G. The vertex-to-clique radius r1 of G is the minimum vertex-to-clique eccentricity among the vertices of G, while the vertex-to-clique diameter d1 of G is the maximum vertex-to-clique eccentricity among the vertices of G. Also the vertex-toclique detour distance, D(u; C) is the length of a longest u¡C path in G. A u ¡ C path of length D(u; C) is called a u ¡ C detour. The vertex-to-clique detour eccentricity eD1(u) of a vertex u in G is the maximum vertex-toclique detour distance from u to a clique C 2 ³ in G. The vertex-to-clique detour radius R1 of G is the minimum vertex-to-clique detour eccentricity among the vertices of G, while the vertex-to-clique detour diameter D1 of G is the maximum vertex-to-clique detour eccentricity among the vertices of G. It is shown that R1 · D1 for every connected graph G and that every two positive integers a and b with 2 · a · b are realizable as the vertex-to-clique detour radius and the vertex-to-clique detour diameter, respectively, of some connected graph. Also it is shown that for any three positive integers a; b; c with 2 · a · b < c, there exists a connected graph G such that r1 = a, R1 = b, R = c and for any three positive integers a; b; c with 2 · a · b < c and a + c · 2b, there exists a connected graph G such that d1 = a, D1 = b, D = c.

Page(s): 45-59

DOI: DOI not available

Published: Journal: Journal of Prime Research in Mathematics, Volume: 12, Issue: 1, Year: 2016

Keywords:

05C12 , AMS Subject Classi¯cation , vertextoclique detour distance , vertextoclique distance , vertextoclique detour center , vertextoclique detour periphery

References:

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