THE TOTAL EDGE FIXING EDGE-TO-EDGE GEODETIC NUMBER OF A GRAPH

Abstract

In this article, we introduce the concept of the total edge fixing edge-to-edge geodetic number ????????????????????????(????)for an edge ????of a graph ????. The total edge fixing edge-to-edge geodetic number of certain classes of graphs including path, cycles, trees, complete graphs are studied. Also it is shown that for every pair of positive integers with 2 ≤ ???? ≤ ????, there exists a graph ???? such that ????????????????(????) = ???? and ????????????????????????(????) = ????,for some edge ????????(????).