The Edge-To-Vertex Triangle Free Detor Distance in Graphs

Abstract

For every connected graph G, the triangle free detour distance D_∆f(u, v) is the length of a longest u- v triangle free path in G, where u, v are the vertices of G.  A u-v triangle free path of length D_∆f(u, v) is called the u-v triangle free detour. In this article, the edge-to-vertex triangle free detour distance is introduced. It is found that the edge -to-vertex triangle free detour distance differs from the edge -to-vertex distance and edge-to-vertex detour distance. The edge-to-vertex triangle free detour distance is found for some standard graphs. Their bounds are determined and their sharpness is checked. Certain general properties satisfied by them are studied.