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Let Ω be a symmetric generating set of a finite group Γ. Assume that (Γ , Ω) be such that Γ = ⟨Ω ⟩ and Ω satisfies the two conditions C 1 : the identity element e ̸ ∈ Ω and C 2 : if a ∈ Ω , then a −1 ∈ Ω. Given (Γ , Ω) satisfying C 1 and C 2 , define a Cayley graph G = Cay(Γ , Ω) with V (G) = Γ and E(G) = {(x, y) a |x, y ∈ Γ , a ∈ Ω and y = xa}. When Γ = Z n = ⟨Ω ⟩, it is called as circulant graph and denoted by Cir (n, Ω). In this paper, we give a survey about the results on dominating sets in Cayley graphs and circulant graphs. c

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Let [Formula: see text] be a commutative ring with identity, [Formula: see text] be a positive integer and [Formula: see text] be the set of all [Formula: see text] matrices over [Formula: see text] For a matrix [Formula: see text] Tr[Formula: see text] is the trace of [Formula: see text] The trace graph of the matrix ring [Formula: see text] denoted by [Formula: see text] is the simple undirected graph with vertex set [Formula: see text][Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if Tr[Formula: see text] The ideal-based trace graph of the matrix ring [Formula: see text] with respect to an ideal [Formula: see text] of [Formula: see text] denoted by [Formula: see text] is the simple undirected graph with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if Tr[Formula: see text] In this paper, we investigate some properties and structure of [Formula: see text] Further, it is proved that both [Formula: see text] and [Formula: see text] are Hamiltonian.

Let H be a connected subgraph of a connected graph G. The H-structure connectivity of the graph G, denoted by κ(G;H), is the minimum cardinality of a minimal set of subgraphs F={H1′,H2′,…,Hm′} in G, such that every H′i∈F is isomorphic to H and removal of F from G will disconnect G. The H-substructure connectivity of the graph G, denoted by κs(G;H), is the minimum cardinality of a minimal set of subgraphs F={J1′,J2′,…,Jm′} in G, such that every Ji′∈F is a connected subgraph of H and removal of F from G will disconnect G. In this paper, we provide the H-structure and the H-substructure connectivity of the circulant graph Cir(n,Ω) where Ω={1,…,k,n−k,…,n−1},1≤k≤⌊n2⌋ and the hypercube Qn for some connected subgraphs H.