For any graph X with the adjacency matrix A, the transition matrix of the continuous-time quantum walk at time t is given by the matrix-valued function H X (t) = e itA . We say that there is perfect state transfer in X from the vertex u to the vertex v at time τ if |H X (τ ) u,v | = 1. It is an important problem to determine whether perfect state transfers can happen on a given family of graphs. In this paper we characterize all the graphs in the Johnson scheme which have this property. Indeed, we show that the Kneser graph K(2k, k) is the only class in the scheme which admits perfect state transfers. We also show that, under some conditions, some of the unions of the graphs in the Johnson scheme admit perfect state transfer.

Let R be a finite commutative ring with 1 = 0. In this article, we study the total graph of R, denoted by τ (R), determine some of its basic graph-theoretical properties, determine when it is Eulerian, and find some conditions under which this graph is isomorphic to Cay(R, Z(R)\{0}). We shall also compute the domination number of τ (R).

A vertex coloring of a graph G is called distinguishing if no non-identity automorphisms of G can preserve it. The distinguishing number of G, denoted by D(G), is the minimum number of colors required for such coloring. The distinguishing threshold of G, denoted by θ(G), is the minimum number k of colors such that every k-coloring of G is distinguishing. In this paper, we study θ(G), find its relation to the cycle structure of the automorphism group and prove that θ(G) = 2 if and only if G is isomorphic to K 2 or K 2 . Moreover, we study graphs that have the distinguishing threshold equal to 3 or more and prove that θ(G) = D(G) if and only if G is asymmetric, K n or K n . Finally, we consider Johnson scheme graphs for their distinguishing number and threshold concludes the paper.

We associate a graph Γ+(R) to a ring R whose vertices are nonzero proper right ideals of R and two vertices I and J are adjacent if I + J = R. Then we try to translate properties of this graph into algebraic properties of R and vice versa. For example, we characterize rings R for which Γ+(R) respectively is connected, complete, planar, complemented or a forest. Also we find the dominating number of Γ+(R).

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