Let [Formula: see text] be a subset of a commutative graded ring [Formula: see text]. The Cayley graph [Formula: see text] is a graph whose vertex set is [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. The Cayley sum graph [Formula: see text] is a graph whose vertex set is [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. Let [Formula: see text] be the set of homogeneous elements and [Formula: see text] be the set of zero-divisors of [Formula: see text]. In this paper, we study [Formula: see text] (total graph) and [Formula: see text]. In particular, if [Formula: see text] is an Artinian graded ring, we show that [Formula: see text] is isomorphic to a Hamming graph and conversely any Hamming graph is isomorphic to a subgraph of [Formula: see text] for some finite graded ring [Formula: see text].
Let Rm = Fq [y] = /〈ym - 1〉, where m|q - 1. In this paper, we obtain the structure of linear and cyclic codes over Rm. Also, we introduce a preserving-orthogonality Gray map from Rm to Fmq. Among the main results, we obtain the exact structure of self-orthogonal cyclic codes over Rm to introduce parameters of quantum codes from cyclic codes over Rm.
Abstract:The unitary Cayley graph on n vertices, Xn, has vertex set Zn, where two vertices a and b are connected by an edge if and only if they differ by a multiplicative unit modulo n, i.e. gcd(ab, n) = 1. A k-regular graph X is Ramanujan if and only if λ(X) ≤ 2 √ k − 1 where λ(X) is the second largest absolute value of the eigenvalues of the adjacency matrix of X. We obtain a complete characterization of the cases in which the complements of unitary Cayley graphXn is a Ramanujan graph.
In a graph G, a vertex dominates itself and its neighbors. A subset S of V is called a dominating set in G if every vertex in V is dominated by at least one vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set. A set S ⊆ V is called a double dominating set of a graph G if every vertex in V is dominated by at least two vertices in S. The minimum cardinality of a double dominating set is called double domination number of G. The connectivity γ(G) of a connected graph G is the minimum number of vertices whose removal results in a disconnected or trivial graph. In this paper, introduced the concept of three domination in graphs. and we obtain an upper bound for the sum of the three domination number and connectivity of a graph and characterize the corresponding extremal graphs.
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