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In this paper, we extend our investigation about the generalized comaximal graph introduced in Biswas et al. (Discrete Math Algorithms Appl 11(1):1950013, 2019a). The generalized comaximal graph is defined as follows: given a finite commutative ring R, the generalized comaximal graph G(R) is an undirected graph with its vertex set comprising elements of R and two distinct vertices u, v are adjacent if and only if there exists a non-zero idempotent e ∈ R such that u R + v R = e R . In this study, we focus on identifying the rings R for which the graph G(R) exhibits planarity. Moreover, we provide a characterization of the class of ring for which G(R) is toroidal, denoted by γ (G (R)) = 1 . Furthermore, we also evaluate the energy of the graph G(R). Finally, we demonstrate that the graph G(R) is always Hamiltonian for any finite commutative ring R. [ABSTRACT FROM AUTHOR]

In this paper we study the problem of constructing non-trivial subtowers and supertowers of recursive towers of function fields over finite fields. [ABSTRACT FROM AUTHOR]

The degree of a vertex or face in a graph on the plane or other orientable surface is the number of incident edges. A face is of type if whenever . We denote the minimum vertex-degree of by . The purpose of our paper is to prove that every triangulation with of the torus, as well as of large enough such a triangulation of any fixed orientable surface of higher genus has a face of one of the types , , , , , or , where all parameters are best possible. [ABSTRACT FROM AUTHOR]

In this paper, we study gapsets and we focus on obtaining information on how the maximum distance between two consecutive elements influences the behaviour of the set. In particular, we prove that the cardinality of the set of gapsets with genus g such that the maximum distance between two consecutive elements is κ is equal to the cardinality of the set of gapsets with genus g + 1 such that the maximum distance between two consecutive elements is κ + 1 , when 2 g ≤ 3 κ . [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with 1. In Biswas et al. (Disc Math Algorithms Appl 11(1):1950013, 2019), we introduced a graph G(R) whose vertices are elements of R and two distinct vertices a, b are adjacent if and only if a R + b R = e R for some nonzero idempotent e in R. Let G ′ (R) be the subgraph of G(R) generated by the non-units of R. In this paper, we characterize those rings R for which the graph G ′ (R) is connected and Eulerian. Also we characterize those rings R for which genus of the graph G ′ (R) is ≤ 2 . Finally, we show that the graph G ′ (R) is a line graph of some graph if and only if R is either a regular ring or a local ring. [ABSTRACT FROM AUTHOR]

Topological constraints have recently been introduced to structural topology optimization by the BESO method. However, for the classical and widely used SIMP-type optimization method, an implicit and continuously changing variable cannot express the topological characteristics directly during the optimization process. This is partly caused by missing well-defined boundaries to compute topological characteristics. To introduce topological constraints into the SIMP-type method, an auxiliary discrete expression of structural boundaries through the volume preservation projection method is used to compute topological characteristics, that is, the genus or number of holes. A topological control methodology based on persistence homology, a numerical calculation idea derived from topological data analysis, is introduced in this paper to implement topological constraints. With the help of the design space progressive restriction method, the proposed methodology shows that for the 2D static minimum compliance optimization problem, the inequality constraints on the number of holes can be satisfied. The effectiveness of the proposed topological control method for the SIMP-type framework is illustrated by several numerical examples. [ABSTRACT FROM AUTHOR]

For a finite noncyclic group G, let Cyc (G) be a set of elements a of G such that ⟨ a , b ⟩ is cyclic for each b of G. The noncyclic graph of G is a graph with the vertex set G \ Cyc (G) , having an edge between two distinct vertices x and y if ⟨ x , y ⟩ is not cyclic. In this paper, we show that, for a fixed nonnegative integer k, there are at most finitely many finite noncyclic groups whose noncyclic graphs have (non)orientable genus k. We also classify the finite noncyclic groups whose noncyclic graphs have (non)orientable genus 1, 2, 3 and 4, respectively. [ABSTRACT FROM AUTHOR]

The concept of Arf numerical semigroup plays a significant role in the study of the equivalence between algebroid branches. Namely, two algebroid branches are said to be equivalent if their Arf closures have the same value semigroup, that is an Arf numerical semigroup. By introducing the concept of Arf good semigroups of N r , it is possible to extend the aforementioned equivalence to the more general context of algebroid curves with r > 1 branches. Arf good semigroups can be completely described by their multiplicity trees that are combinatorial objects whose study is independent from the ring theory context. In this paper we give an algorithm for the computation of all Arf numerical semigroups with a given genus. Moreover, we generalize the concept of genus of a numerical semigroup to good semigroups of N r and we give a procedure to calculate all Arf good semigroups of N r with a given genus. [ABSTRACT FROM AUTHOR]

In this paper, we study the set L(m,r)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathscr {L}(m,r)$$\end{document} of all numerical semigroups with multiplicity m and ratio r. In particular, we present some algorithms that compute all the elements of L(m,r)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \mathscr {L}(m,r)$$\end{document} with a given genus or with a given Frobenius number. [ABSTRACT FROM AUTHOR]

The aim of this paper is to characterize the first graded Hochschild cohomology of a hereditary algebra whose Gabriel quiver is admitted to have oriented cycles. The interesting conclusion we have obtained shows that the standard basis of the first graded Hochschild cohomology depends on the genus of a quiver as a topological object. In this paper, we overcome the limitation of the classical Hochschild cohomology for hereditary algebra where the Gabriel quiver is assumed to be acyclic. As preparation, we first investigate the graded differential operators on a path algebra and the associated graded Lie algebra. [ABSTRACT FROM AUTHOR]

Let R be a finite commutative ring with nonzero identity and denote its Jacobson radical by J( R). The Jacobson graph of R is the graph in which the vertex set is $$R\setminus J(R)$$ , and two distinct vertices x and y are adjacent if and only if $$1-xy$$ is not a unit in R. In this paper, up to isomorphism, we classify the rings R whose Jacobson graphs are toroidal. [ABSTRACT FROM AUTHOR]

In this paper we study properties of topological RNA structures, i.e. RNA contact structures with cross-serial interactions that are filtered by their topological genus. RNA secondary structures within this framework are topological structures having genus zero. We derive a new bivariate generating function whose singular expansion allows us to analyze the distributions of arcs, stacks, hairpin- , interior- and multi-loops. We then extend this analysis to H-type pseudoknots, kissing hairpins as well as 3-knots and compute their respective expectation values. Finally we discuss our results and put them into context with data obtained by uniform sampling structures of fixed genus. [ABSTRACT FROM AUTHOR]

A graph G is said to be equimatchable if every matching in G extends to (i.e., is a subset of) a maximum matching in G. In an earlier paper with Saito, the authors showed that there are only a finite number of 3-connected equimatchable planar graphs. In the present paper, this result is extended by showing that in a surface of any fixed genus (orientable or non-orientable), there are only a finite number of 3-connected equimatchable graphs having a minimal embedding of representativity at least three. (In fact, if the graphs considered are non-bipartite, the representativity three hypothesis may be dropped.) The proof makes use of the Gallai-Edmonds decomposition theorem for matchings. [ABSTRACT FROM AUTHOR]

In the paper, we prove that for every integer n ≥ 1, there exists a Petersen power P with nonorientable genus and Euler genus precisely n, which improves the upper bound of Mohar and Vodopivec's result [ J. Graph Theory, 67, 1-8 (2011)] that for every integer k (2 ≤ k ≤ n−1), a Petersen power P exists with nonorientable genus and Euler genus precisely k. [ABSTRACT FROM AUTHOR]

Let $$p,q\in \mathbb {N}$$ be relatively prime integers with $$1

In this paper we present an algorithm for computing the set of saturated numerical semigroups of a given genus. We see how the set of saturated numerical semigroups can be arranged in a tree rooted in $\mathbb{N}$ and we describe the sons of any vertex of this tree. [ABSTRACT FROM AUTHOR]

This paper extends the methodology for the construction of odd polyhedral links. Building blocks are odd chain tangles, each of which consists of finitely many $$2n+1$$-twist tangles for any nonnegative integer $$n$$. For any polyhedral graph $$G$$, replacing each edge with an odd chain tangle results in an infinite collection of odd polyhedral links. The relationship between the HOMFLY polynomials of these odd links and the $$Q^{d}$$-polynomial of $$G$$ is established. It leads to the determination of the span of the HOMFLY polynomial, the bound on the braid index and the genus of each odd link. Our results show that these indices depend not only on the building blocks but also on the graph $$G.$$ [ABSTRACT FROM AUTHOR]

In this paper we present a new approach to construct the set of numerical semigroups with a fixed genus. Our methodology is based on the construction of the set of numerical semigroups with fixed Frobenius number and genus. An equivalence relation is given over this set and a tree structure is defined for each equivalence class. We also provide a more efficient algorithm based on the translation of a numerical semigroup to its so-called Kunz-coordinates vector. [ABSTRACT FROM AUTHOR]

On the basis of the joint tree model initiated in Liu (The non-orientable maximum genus of a graph (in Chinese), Scieulia Sinica, Special Issue on Math. I 191-201, ) and comprehensively described in Liu (Theory of Polyhedra (in English), Science Press, Beijing, ), this paper provides the numbers of topologically non-equivalent orientable embeddings of a new type of 3-regular graphs by genus via classifying the associate polyhegons. [ABSTRACT FROM AUTHOR]

Two cellular embeddings i: G → S and j: G → S of a connected graph G into a closed orientable surface S are equivalent if there is an orientation-preserving surface homeomorphism h: S → S such that hi = j. The genus polynomial of a graph G is defined by where a is the number of equivalence classes of embeddings of G into the orientable surface S with g genera. In this paper, we compute the genus polynomial of a graph obtained from a cycle by replacing each edge by two multiple edges. [ABSTRACT FROM AUTHOR]

Let a1,..., a n be relatively prime positive integers, and let S be the semigroup consisting of all non-negative integer linear combinations of a1,..., a n. In this paper, we focus our attention on AA-semigroups, that is semigroups being generated by almost arithmetic progressions. After some general considerations, we give a characterization of the symmetric AA-semigroups. We also present an efficient method to determine an Apéry set and the Hilbert series of an AA-semigroup. [ABSTRACT FROM AUTHOR]

A curve defined over a finite field is maximal or minimal according to whether the number of rational points attains the upper or the lower bound in Hasse-Weil’s theorem, respectively. In the study of maximal curves a fundamental role is played by an invariant linear system introduced by Rück and Stichtenoth in [6]. In this paper we define an analogous invariant system for minimal curves, and we compute its orders and its Weierstrass points. In the last section we treat the case of curves having genus three in characteristic two. [ABSTRACT FROM AUTHOR]

In this paper, we study the following elliptic boundary value problem: { − Δ u + V (x) u = f (x , u) , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where Ω ⊂ R N is a bounded domain with smooth boundary ∂Ω, and f is allowed to be sign-changing and is of sublinear growth near infinity in u. For both cases that V ∈ L N / 2 (Ω) with N ≥ 3 and that V ∈ C (Ω , R) with inf Ω V (x) > − ∞ , we establish a sequence of nontrivial solutions converging to zero for above equation via a new critical point theorem. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with a nonzero identity element. For a natural number n, we associate a simple graph, denoted by ΓRn, with Rn\{0} as the vertex set and two distinct vertices X and Y in Rn being adjacent if and only if there exists an n×n lower triangular matrix A over R whose entries on the main diagonal are nonzero and one of the entries on the main diagonal is regular such that XTAY=0 or YTAX=0, where, for a matrix B,BT is the matrix transpose of B. If n=1, then ΓRn is isomorphic to the zero divisor graph Γ(R), and so ΓRn is a generalization of Γ(R) which is called a generalized zero divisor graph of R. In this paper, we study some basic properties of ΓRn. We also determine all isomorphic classes of finite commutative rings whose generalized zero divisor graphs have genus at most three. [ABSTRACT FROM AUTHOR]