In this paper, we introduce the concept of edge-to-vertex strong geodetic number (G) of a connected graph with at least 3 vertices. Some general properties satisfied by these concepts are studied. The edge-to-vertex strong geodetic number of certain classes of graphs is determined. It is proved that for each pair of integers k and with , there exists a connected graph of order and size with ( ) . It is shown that for positive integers and with , there exists a connected graph with , ( ) . Connected graphs of size with edge-to-vertex strong geodetic number – – are characterized.