1. Generalization of edge general position problemPaul Manuel, R. Prabha, Sandi Klavžar, 2025, original scientific article Abstract: The edge geodesic cover problem of a graph G is to find a smallest number of geodesics that cover the edge set of G. The edge k-general position problem is introduced as the problem to find a largest set S of edges of G such that at most k-1 edges of S lie on a common geodesic. We show that these are dual min-max problems and connect them to an edge geodesic partition problem. Using these connections, exact values of the edge k-general position number is determined for different values of k and for various networks including torus networks, hypercubes, and Benes networks.
Keywords: general position set, edge geodesic cover problem, edge k-general position problem, torus network, hypercube, Benes network
Published in RUP: 03.11.2025; Views: 251; Downloads: 1
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2. Horadam cubesLuka Podrug, 2025, original scientific article Abstract: We define and investigate a new three-parameter family of graphs that further generalizes the Fibonacci and metallic cubes. Namely, the number of vertices in this family of graphs satisfies Horadam recurrence, a linear recurrence of second order with constant coefficients. It is shown that the new family preserves many appealing and useful properties of the Fibonacci and metallic cubes. In particular, we present recursive decomposition and decomposition into grids and explore some metric and enumerative properties such as the number of edges, distribution of degrees, and cube polynomials. We also investigate the existence of Hamiltonian paths and cycles.
Keywords: hypercube, Fibonacci cube, metallic cube, Horadam's recurrence
Published in RUP: 22.10.2025; Views: 286; Downloads: 4
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3. The spanning laceability on the faulty bipartite hypercube-like networksCheng-Kuan Lin, Yuan-Hsiang Teng, Jimmy J. M. Tan, Lih-Hsing Hsu, Dragan Marušič, 2013, original scientific article Keywords: Hamiltonian, Hamiltonian laceable, Hypercube networks, Hypercube-like network, Spanning laceability
Published in RUP: 15.10.2013; Views: 4015; Downloads: 89
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4. Leonard triples and hypercubesŠtefko Miklavič, 2007, original scientific article Abstract: Let ▫$V$▫ denote a vector space over ▫$\mathbb{C}$▫ with finite positive dimension. By a Leonard triple on ▫$V$▫ we mean an ordered triple of linear operators on ▫$V$▫ such that for each of these operators there exists a basis of ▫$V$▫ with respect to which the matrix representing that operator is diagonal and the matrices representing the other two operators are irreducible tridiagonal. Let ▫$D$▫ denote a positive integer and let ▫${\mathcal{Q}}_D$▫ denote the graph of the ▫$D$▫-dimensional hypercube. Let ▫$X$ denote the vertex set of ▫${\mathcal{Q}}_D$▫ and let ▫$A \in {\mathrm{Mat}}_X ({\mathbb{C}})$▫ denote the adjacency matrix of ▫${\mathcal{Q}}_D$▫. Fix ▫$x \in X$▫ and let ▫$A^\ast \in {\mathrm{Mat}}_X({\mathbb{C}})$▫ denote the corresponding dual adjacency matrix. Let ▫$T$▫ denote the subalgebra of ▫${\mathrm{Mat}}_X({\mathbb{C}})$ generated by ▫$A,A^\ast$▫. We refer to ▫$T$▫ as the Terwilliger algebra of ▫${\mathcal{Q}}_D$▫ with respect to ▫$x$▫. The matrices ▫$A$▫ and ▫$A^\ast$▫ are related by the fact that ▫$2iA = A^\ast A^\varepsilon - A^\varepsilon A^\ast$▫ and ▫$2iA^\ast = A^\varepsilon A - AA^\varepsilon$▫, where ▫$2iA^\varepsilon = AA^\ast - A^\ast A$▫ and ▫$i^2 = -1$▫. We show that the triple ▫$A$▫, ▫$A^\ast$▫, ▫$A^\varepsilon$▫ acts on each irreducible ▫$T$▫-module as a Leonard triple. We give a detailed description of these Leonard triples.
Keywords: mathematics, graph theory, Leonard triple, distance-regular graph, hypercube, Terwilliger algebra
Published in RUP: 15.10.2013; Views: 9409; Downloads: 127
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