فهرست مطالب
Transactions on Combinatorics
Volume:7 Issue: 4, Dec 2018
- تاریخ انتشار: 1397/09/21
- تعداد عناوین: 5
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Pages 1-6It is shown that the certain combinatorial structures called stopping sets have the important role in analysis of iterative decoding. In this paper, the number of minimum stopping sets of a product code is determined by the number of the minimum stopping sets of the corresponding component codes. As an example, the number of minimum stopping sets of the r-dimensional SPC product code is computed.Keywords: Stopping set, Stopping distance, Product code
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Pages 7-10Alspach et al. conjectured that every quartic Cayley graph on an even solvable group is 1-factorizable. In this paper, we verify this conjecture for quartic Cayley graphs on supersolvable groups of even order.Keywords: Cayley graph, 1-factorization, supersolvable group
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Pages 11-24The degree resistance distance of a graph G is defined as DR(G)=∑i<j(d(vi)+d(vj))R(vi,vj), where d(vi) is the degree of the vertex vi, and R(vi,vj) is the resistance distance between the vertices vi and vj . Here we characterize the extremal graphs with respect to degree resistance distance among trees with given diameter, number of pendent vertices, independence number, covering number, and maximum degree, respectively.Keywords: Trees, Degree resistance distance, Diameter, Covering number
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Pages 25-42We introduce new refinements of the Bell, factorial, and unsigned Stirling numbers of the first and second kind that unite the derangement, involution, associated factorial, associated Bell, incomplete Stirling, restricted factorial, restricted Bell, and r-derangement numbers (and probably more!). By combining methods from analytic combinatorics, umbral calculus, and probability theory, we derive several recurrence relations and closed form expressions for these numbers. By specializing our results to the classical case, we recover explicit formulae for the Bell and Stirling numbers as sums over compositions.Keywords: Bell numbers, Stirling numbers
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Pages 43-57Let R be an associative ring with identity and Z∗(R) be its set of non-zero zero-divisors. Zero-divisor graphs of rings are well represented in the literature of commutative and non-commutative rings. The directed zero-divisor graph of R, denoted by Γ(R), is the directed graph whose vertices are the set of non-zero zero-divisors of R and for distinct non-zero zero-divisors x,y, x→y is an directed edge if and only if xy=0. In this paper, we connect some graph-theoretic concepts with algebraic notions, and investigate the interplay between the ring-theoretical properties of a skew power series ring R[[x;α]] and the graph-theoretical properties of its directed zero-divisor graph Γ(R[[x;α]]). In doing so, we give a characterization of the possible diameters of Γ(R[[x;α]]) in terms of the diameter of Γ(R), when the base ring R is reversible and right Noetherian with an α-condition, namely α-compatible property. We also provide many examples for showing the necessity of our assumptionsKeywords: Zero-divisor graphs, Diameter, Reversible rings, Noetherian rings, Skew power series rings