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Transactions on Combinatorics - Volume:7 Issue: 4, Dec 2018

Transactions on Combinatorics
Volume:7 Issue: 4, Dec 2018

  • تاریخ انتشار: 1397/09/21
  • تعداد عناوین: 5
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  • Morteza Hivadi *, Akbar Zare Chavoshi Pages 1-6
    It 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
  • Milad Ahanjideh, Ali Iranmanesh * Pages 7-10
    Alspach 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
  • Fangguo He *, Xinnong Jiang Pages 11-24
    The 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
  • Tanay Wakhare * Pages 25-42
    ‎‎We 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
  • Ebrahim Hashemi *, Marzieh Yazdanfar, Abdollah Alhevaz Pages 43-57
    ‎Let 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 assumptions
    Keywords: ‎Zero-divisor graphs‎, ‎Diameter‎, ‎Reversible rings‎, ‎Noetherian rings‎, ‎Skew power series rings