Aug 2026· Utilitas mathematica· Vol 128, pp. 185-200· 0 citations· 11 references
Mathematics
Abstract
We introduce and study a new family of circulant digraphs associated with the cyclic group \({\mathbb Z}_N\), obtained by restricting admissible combinations of two generators \(a\) and \(b\) to three coordinate sectors. The resulting distance-like function differs from the standard directed distance in circulant digraphs and gives rise to new geometric and combinatorial phenomena. Using planar lattice representations and periodic tessellations, we analyze the growth of reachable sets and derive Moore-type upper bounds for the corresponding order/diameter problem. We construct explicit infinite families of three-quarters circulant structures with the prescribed diameter and provide lattice-based methods for determining admissible generator pairs. Separate constructions are obtained for even and odd diameters. In addition, computational experiments for small and moderate orders suggest improved families for even diameters and motivate a conjectural asymptotic formula for the maximum attainable order. The paper highlights the interplay between constrained lattice representations, periodic tilings, and extremal problems for circulant networks.
The derived set of multiple zeta-star values is the half-line $[1,+\infty)$. In this paper we study the corresponding two-dimensional problem for multiple star polylogarithms on the unit circle. We first prove that every shifted multiple star polylogarithm is the generating function of a completely monotone sequence an...
We generalize intersection numbers for combinatorial designs to designs in finite meet-semilattices satisfying suitable regularity conditions. While designs in regular semilattices go back to Delsarte, our regularity assumptions are weaker than his and need not give rise to an association scheme. In this framework, we...
We introduce Picard Farey symbols for finite-index subgroups of \(G=\PSL_2(\ZZ[i])\), using the Gaussian Farey tessellation of \(\HH^3\) by ideal octahedra. A symbol records finitely many Gaussian-rational octahedral occurrences, their local stabilizers, and ordered cooriented face pairings. We prove that a valid marke...
Unit interval graphs admit a classical Catalan encoding by area sequences, or equivalently by Dyck paths. We introduce a new statistic on these graphs, called peak-nesting, defined as the largest number of peak-cliques containing a common vertex. Through the correspondence with area sequences, peak-nesting also defines...
Breakpoint graphs originate in comparative genomics, where their alternating cycles encode relationships between genomes. We study a constrained class of three-colored breakpoint graphs associated with permutations and develop cycle-refined generating functions for two extremal families. These families have a natural t...
M. Alekseyev, Joseph T. Iosue, Adam Ehrenberg et al.· 1 citation· ⚡1
The restricted augmented Zarankiewicz number \(z_L(m,n)\) yields core combinatorial lower bounds for the maximal SOS rank of biquadratic forms. All previously known infinite admissible graph families rely on \(K_{4t}\) incidence graphs, attaining an asymptotic relative gap limit of \(1/4\). This work develops a new inf...
Han-Xin Liu, Yi-Sheng Song· 1 citation
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