Journal of Prime Research in Mathematics

# Clique-to-vertex detour distance in graphs

I. Keerthi Asir
Department of Mathematics, St. Xavier’s College (Autonomous), Palayamkottai –
$$^{1}$$Corresponding Author: asirsxc@gmail.com
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 $$d_2 = a$$, $$D_2 = b$$.