Let Γ be a simple connected undirected graph with vertex set VΓ and edge set EΓ. The metric dimension of a graph Γ is the least number of vertices in a set with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. For an ordered subset W=w1,w2,…,wk of vertices in a graph Γ and a vertex v of Γ, the metric representation of v with respect to W is the k-vector rvW=dv,w1,dv,w2,…,dv,wk. If every pair of distinct vertices of Γ have different metric representations, then the ordered set W is called a resolving set of Γ. It is known that the problem of computing this invariant is NP-hard. In this paper, we consider the problem of determining the cardinality ψΓ of minimal doubly resolving sets of Γ and the strong metric dimension for the jellyfish graph JFGn,m and the cocktail party graph CPk+1.

Let G be a finite, connected graph of order of, at least, 2 with vertex set VG and edge set EG. A set S of vertices of the graph G is a doubly resolving set for G if every two distinct vertices of G are doubly resolved by some two vertices of S. The minimal doubly resolving set of vertices of graph G is a doubly resolving set with minimum cardinality and is denoted by ψG. In this paper, first, we construct a class of graphs of order 2n+Σr=1k−2nmr, denoted by LSGn,m,k, and call these graphs as the layer Sun graphs with parameters n, m, and k. Moreover, we compute minimal doubly resolving sets and the strong metric dimension of the layer Sun graph LSGn,m,k and the line graph of the layer Sun graph LSGn,m,k.

Let n and k be integers with n > k ≥ 1 and [n] = {1, 2, ..., n}. The bipartite Kneser graph H(n, k) is the graph with the all k-element and all (n − k)-element subsets of [n] as vertices, and there is an edge between any two vertices, when one is a subset of the other. In this paper, we show that H(n, k) is an arc-transitive graph. Also, we show that H(n, 1) is a distance-transitive Cayley graph. Finally, we determine the automorphism group of the graph H(n, 1) and show that Aut(H(n, 1)) ∼ = Sym([n]) × Z 2 , where Z 2 is the cyclic group of order 2. Moreover, we pose some open problems about the automorphism group of the bipartite Kneser graph H(n, k).

Let Γ = Cay(Z n , S k ) be the Cayley graph on the cyclic additive group Z n (n ≥ 4), where] − 1. In this paper, we will show that χ(Γ) = ω(Γ) = k + 1 if and only if k + 1|n. Also, we will show that if n is an even integer and k = n 2 − 1 then Aut(Γ) ∼ = Z 2 wr I Sym(k + 1) where I = {1, . . . , k + 1} and in this case, we show that Γ is an integral graph.

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