‎In this paper we give a characterization of unmixed tripartite‎ ‎graphs under certain conditions which is a generalization of a‎ ‎result of Villarreal on bipartite graphs‎. ‎For bipartite graphs two‎ ‎different characterizations were given by Ravindra and Villarreal‎. ‎We show that these two characterizations imply each other‎.

Let q be a prime power, Fq be the field of q elements, and k;m be positive integers. A bipartite graph G 1⁄4 Gqðk;mÞ is defined as follows. The vertex set of G is a union of two copies P and L of two-dimensional vector spaces over Fq, with two vertices ðp1;p2Þ 2 P and 1⁄2 l1; l2 2 L being adjacent if and only if p2 þ l2 1⁄4 p 1 l 1 . We prove that graphs Gqðk ;mÞ and Gq0 ðk 0;m0Þ are isomorphic...

Let q be a prime power, Fq be the field of q elements, and k, m be positive integers. A bipartite graph G = Gq(g, h, k,m) is defined as follows. The vertex set of G is a union of two copies P and L of two-dimensional vector spaces over Fq, with two vertices (p1, p2) ∈ P and [ l1, l2 ] ∈ L being adjacent if and only if p2 + l2 = p1l 1 . We prove that graphs Gq(k, m) and Gq′(k,m) are isomorphic i...