Contributions to Discrete Mathematics (E-Journal, University of Calgary)
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A method to determine algebraically integral Cayley digraphs on finite abelian group
Get PDFResearchers in the past have studied eigenvalues of Cayley digraphs or graphs. We are interested in characterizing Cayley digraphs on a finite commutative group whose eigenvalues are algebraic integers in a given number field We succeed in finding a method to do so. The number of such Cayley digraphs are computed
Partitioning the array into restrictions of circles
No full textWe show that there is a unique way to partition a array of lattice points into restrictions of five circles. This result is extended to the array, and used to show the optimality of a six-circle solution for the array
3-uniform hypergraphs: modular decomposition and realization by tournaments
No full textLet be a 3-uniform hypergraph. A tournament defined on is a realization of if the edges of are exactly the 3-element subsets of that induce 3-cycles. We characterize the 3-uniform hypergraphs that admit realizations by using a suitable modular decomposition
Construction of strongly regular graphs having an automorphism group of composite order
Get PDFIn this paper we outline a method for constructing strongly regular graphs from orbit matrices admitting an automorphism group of composite order. In 2011, C. Lam and M. Behbahani introduced the concept of orbit matrices of strongly regular graphs and developed an algorithm for the construction of orbit matrices of strongly regular graphs with a presumed automorphism group of prime order, and construction of corresponding strongly regular graphs. The method of constructing strongly regular graphs developed and employed in this paper is a generalization of that developed by C. Lam and M. Behbahani. Using this method we classify SRGs with parameters (49,18,7,6) having an automorphism group of order six. Eleven of the SRGs with parameters (49,18,7,6) constructed in that way are new. We obtain an additional 385 new SRGs(49,18,7,6) by switching. Comparing the constructed graphs with previously known SRGs with these parameters, we conclude that up to isomorphism there are at least 727 SRGs with parameters (49,18,7,6). Further, we show that there are no SRGs with parameters (99,14,1,2) having an automorphism group of order six or nine, i.e. we rule out automorphism groups isomorphic to , , , or
The 2-tuple dominating independent number of a random graph
Get PDFIn this note, we show that 2-tuple dominating independent number of the Erd\H{o}s--R\\u27{e}nyi graph a.a.s.~has a two-point concentration when is a constant
On parity and recurrences for certain partition functions
Get PDFIn this paper, parity and recurrence formulas for some partition functions are given. In particular, a new recurrence for the number of partitions of a positive integer into distinct parts is derived and some identities reminiscent of Legendre\u27s partition-theoretic interpretation of Euler\u27s pentagonal numbers theorem are obtained
Decomposition of complete tripartite graphs into cycles and paths of length three
Get PDFLet and denote a cycle and a path on vertices, respectively. In this paper, we obtain necessary and sufficient conditions for the decomposition of into copies of and copies of for all possible values of ,
Binomial transforms of the balancing and Lucas-balancing polynomials
Get PDFIn this study, we define the binomial transforms of balancing and Lucas-balancing polynomials. Also, the generating functions, Binet formulas, and summations of these transforms are found by recurrence relations. Furthermore, we establish the relations between these transforms by deriving new formulas. Finally, we obtain the Catalan and Cassini formulas for these transforms
Normality of one-matching semi-Cayley graphs over finite abelian groups with maximum degree 3
Get PDFA graph is said to be a semi-Cayley graph over a group if it admits as a semiregular automorphism group with two orbits of equal size. We say that is normal if is a normal subgroup of . We prove that every connected intransitive one-matching semi-Cayley graph, with maximum degree three, over a finite abelian group is normal and characterize all such nonnormal graphs
A complete solution to the spectrum problem for graphs with six vertices and up to nine edges
Get PDFLet be a graph. A -design of order is a decomposition of the complete graph into disjoint copies of . The existence problem of graph designs has been completely solved for all graphs with up to five vertices, and all graphs with six vertices and up to seven edges; and almost completely solved for all graphs with six vertices and eight edges leaving two cases of order 32 unsettled. Up to isomorphism there are 20 graphs with six vertices and nine edges (and no isolated vertex). The spectrum problem has been solved completely for 11 of these graphs, and partially for 2 of these graphs. In this article, the two missing graph designs for the six-vertex eight-edge graphs are constructed, and a complete solution to the spectrum problem for the six-vertex nine-edge graphs is given; completing the spectrum problem for all graphs with six vertices and up to nine edges