### 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.

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DOI: http://dx.doi.org/10.23755/rm.v44i0.894

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