**Journal of Prime Research in Mathematics**

Vol. 1 (2015), Issue 1, pp. 42 – 54

ISSN: 1817-3462 (Online) 1818-5495 (Print)

ISSN: 1817-3462 (Online) 1818-5495 (Print)

# Clique-to-vertex detour distance in graphs

**I. Keerthi Asir
**Department of Mathematics, St. Xavier’s College (Autonomous), Palayamkottai –

627 002, Tamil Nadu, India.

**Head, Department of Mathematics, St. Xavier’s College (Autonomous), Palayamkottai –**

S. Athisayanathan

S. Athisayanathan

627 002, Tamil Nadu, India.

\(^{1}\)Corresponding Author: asirsxc@gmail.com

Copyright © 2015 I. Keerthi Asir, S. Athisayanathan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Published:**December, 2015.

### Abstract

Let \(C\) be a clique and \(v\) a vertex in a connected graph \(G\). A clique-to-vertex \(C − v\) path \(P\) is a \(u − v\) path, where u is a vertex in \(C\) such that \(P\) contains no vertices of \(C\) other than \(u\). The clique-to-vertex distance, \(d(C, v)\) is the length of a smallest \(C − v\) path in \(G\). A \(C − v\) path of length \(d(C, v)\) is called a \(C − v\) geodesic. The clique-to-vertex eccentricity \(e_2(C)\) of a clique \(C\) in G is the maximum clique-to-vertex distance from \(C\) to a vertex \(v ∈ V\) in \(G\). The clique-to-vertex radius \(r_2\) of \(G\) is the minimum clique-to-vertex eccentricity among the cliques of \(G\), while the clique-to-vertex diameter \(d-2\) of \(G\) is the maximum cliqueto-vertex eccentricity among the cliques of \(G\). Also The clique-to-vertex detour distance, \(D(C, v)\) is the length of a longest \(C − v\) path in \(G\). A \(C −v\) path of length \(D(C, v)\) is called a (C −v\) detour. The clique-to-vertex detour eccentricity \(e_{D2}(C)\) of a clique \(C\) in \(G\) is the maximum clique-tovertex detour distance from \(C\) to a vertex \(v ∈ V\) in (G\). The clique-to-vertex detour radius \(R_2\) of \(G\) is the minimum clique-to-vertex detour eccentricity among the cliques of \(G\), while the clique-to-vertex detour diameter \(D_2\) of \(G\) is the maximum clique-to-vertex detour eccentricity among the cliques of \(G\). It is shown that \(R_2 ≤ D_2\) for every connected graph \(G\) and that every two positive integers a and b with \(2 ≤ a ≤ b\) are realizable as the clique-tovertex detour radius and the clique-to-vertex detour diameter respectively of some connected graph. Also it is shown that for any two positive integers a and b with \(2 ≤ a ≤ b\), there exists a connected graph \(G\) such that \(r_2 = a\), \(R_2 = b\) and it is shown that for any two positive integers a and b with \(2 ≤ a ≤ b\), there exists a connected graph \(G\) such that \(d_2 = a\), \(D_2 = b\).

#### Keywords:

Clique-to-vertex detour distance, clique-to-vertex detour center, clique-to-vertex detour periphery.