In this paper

we proved that the join of two simple graphs G1 and G2

denoted G1∨G2

is a self-centered graph if and only G1 and G2 are both complete graphs or they satisfy △(Gi)≤v(Gi)-2

i=1

2; where △(G)means th greates degree of vertices in G

and v(G) denotes the number of vertex of G. We shown that the composition of simple connected graph G_1and simple graph G2 is a self-centered graph if and only if (i) G1 and G2 are both complete graphs; or (ⅱ) there is a vextex in G1 with degree v(G1)—1

and △(G2)≤v(G2)-2; or (ⅲ) G1 is a non—complete self—centered graph. Then we shown that the Cartesian product of arbitrary n simple connected graphs G1 (1≤i≤n. n≥2) is self-centered if and only if G1 is self-cen-tered for each i (1≤i≤n).The last examples demonstrated that resulting graphs of other graphical operations need not be self-centered.