The fuzzy coloring of a fuzzy graph was defined by the authors in Eslahchi and Onagh (2004). In this paper we define the chromatic fuzzy sum and strength of fuzzy graph. Some properties of these concepts are studied. It is shown that there exists an upper (a lower) bound for the chromatic fuzzy sum of a fuzzy graph.

Connected cubic graphs of order 10p 3 which admit an automorphism group acting semisymmetrically are investigated. We prove that every connected cubic edge-transitive graph of order 10p 3 is vertex-transitive, where p is a prime.

The harmonic index of a graph G is defined as the sum of the weights 2 deg G ðuÞþdeg G ðvÞ of all edges uv of G, where deg G ðuÞ denotes the degree of a vertex u in G. In this paper, we investigate the harmonic index of Cartesian, lexicographic, tensor, strong, corona and edge corona product of two connected graphs.

A graph is called symmetric if its automorphism group acts transitively on its arc set. In this paper, we classify all connected cubic symmetric graphs of order 16 p 2 for each prime p.

The F-sum G 1 + F G 2 of two connected graphs G 1 and G 2 is a graph with vertex set (V (G 1) ∪ E(G 1)) × V (G 2) in which two vertices (u 1 , v 1) and (u 2 , v 2) of G 1 + F G 2 are adjacent if and only if [u 1 = u 2 ∈ V (G 1) and v 1 v 2 ∈ E(G 2)] or [v 1 = v 2 and u 1 u 2 ∈ E(F (G))] [M. Eliasi and B. Taeri, Four new sums of graphs and their Wiener indices, Discrete Appl. Math. 157 (2009), 794-803]. In this paper, we investigate the harmonic index of edge-semitotal graphs, total graphs and F-sum of graphs, where F = Q or T .