This paper characterizes the convex subgraphs of the join and the composition of graphs. Moreover, we establish the convex subgraph polynomials of graphs resulting from the two graph operations.

A subgraph H of a graph G is said to be convex if for every pair of vertices in H, any u-v geodesic in G lies entirely in H. We define the convex subgraph polynomial of G as the genereting function of the sequence c i (G) ∞ i=1 where c i (G) is the number of convex subgraphs of G of order i. In this paper, we established some properties of this polynomial and relate graph theoretic concepts with algebraic properties of this polynomial. Moreover, we generated the explicit forms of the convex subgraph polynomials of some special graphs. Lastly, we have shown that the sum of the zeros of this polynomial is the negative of the the coefficient of x |V (G)|−1 and the product of the zeros is (−1) |V (G)| .

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