CLIQUE-TO-CLIQUE TRIANGLE FREE DETOUR DISTANCE IN GRAPHS

This paper introduces the clique-to-clique C - C' triangle free path, the clique-to-clique triangle free detour distance D-Delta f(C,C'), the clique-to-clique triangle free detour eccentricity e(Delta f3)(C), the clique-to-clique triangle free detour radius R-Delta f3 (C), and the clique-to-clique triangle free detour diameter D-Delta f3 of a connected graph G, where C and C' are any two cliques in G. These parameters are determined for some standard graphs. It is shown that every two positive integers a and b with 2 <= a <= b are realizable as the clique-to-clique triangle free detour radius and the clique-to-clique triangle free detour diameter, respectively, of some connected graph. Further it is shown that any three positive integers a, b, c with 3 <= a <= b <= c are realizable as the clique-to-clique radius, the clique-to-clique triangle free detour radius, and the clique-to-clique detour radius, respectively, of some connected graph and also any three positive integers a, b, c with 4 <= a <= b <= c are realizable as the clique-to-clique diameter, the clique-to-clique triangle free detour diamater, and the clique-to-clique detour diameter, respectively, of some connected graph. The clique-to-clique triangle free detour center C-Delta f3(G) and the clique-to-clique triangle free detour periphery P-Delta f3(G) are introduced. It is shown that the clique-to-clique triangle free detour center for a connected graph does not lie in a single block of G.