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Abstract: Abstract A graph is edge-transitive if its automorphism group acts transitively on the set of edges of the graph. In this paper, we classify hexavalent edge-transitive graphs of order \(3p^2\) for each prime p. PubDate: 2023-12-01
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Abstract: Abstract Suppose that A is a finite, nonempty subset of a cyclic group of either infinite or prime order. We show that if the difference set \(A-A\) is “not too large”, then there is a nonzero group element with at least as many as \((2+o(1)) A ^2/ A-A \) representations as a difference of two elements of A; that is, the second largest number of representations is, essentially, twice the average. Here the coefficient 2 is best possible. We also prove continuous and multidimensional versions of this result, and obtain similar results for sufficiently dense subsets of an arbitrary abelian group. PubDate: 2023-12-01
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Abstract: Abstract We present a method which in principal allows to characterise all integral circulant graphs with multiplicative divisor set having a spectrum, i.e. the set of distinct eigenvalues, of any given size. We shall exemplify the method for spectra of up to four eigenvalues, also reproving some known results for three eigenvalues along the way. In particular we show that given any integral circulant graph of arbitrary order n with multiplicative divisor set and precisely four distinct eigenvalues, n necessarily is either a prime power or the product of two prime powers with explicitly given simply structured divisor set and set of eigenvalues in both cases. PubDate: 2023-12-01
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Abstract: Abstract We extend the concept of graph representations modulo integers introduced by Erdös and Evans to graph representations over finite rings and generalize it to representations of signed graphs. We introduce several representation numbers and product dimensions of graphs and signed graphs and compute these quantities for a few special classes of signed graphs. PubDate: 2023-12-01
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Abstract: Abstract Using the Fourier transforms at irreducible unitary representations, we introduce the plateaued functions on finite nonabelian groups, which are a generalization of plateaued functions on finite abelian groups as well as bent functions on finite (abelian or nonabelian) groups. As irreducible unitary representations of finite nonabelian groups are much more complicated than characters of finite abelian groups, plateaued functions on finite nonabelian groups are much more intricate than those functions on finite abelian groups. We will discuss the characterizations of plateaued functions on arbitrary finite groups, and study their relations with partial geometric difference sets. We will also discuss the constructions of plateaued functions on finite nonabelian groups. By using the Clifford Theorem in the representation theory of finite groups, we will construct the plateaued functions on a class of finite nonabelian groups with abelian normal subgroups. PubDate: 2023-12-01
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Abstract: Abstract We study the homological algebra of edge ideals of Erdős–Rényi random graphs. These random graphs are generated by deleting edges of a complete graph on n vertices independently of each other with probability \(1-p\) . We focus on some aspects of these random edge ideals—linear resolution, unmixedness and algebraic invariants like the Castelnuovo–Mumford regularity, projective dimension and depth. We first show a double phase transition for existence of linear presentation and resolution and determine the critical windows as well. As a consequence, we obtain that except for a very specific choice of parameters (i.e., \(n,p:= p(n)\) ), with high probability, a random edge ideal has linear presentation if and only if it has linear resolution. This shows certain conjectures hold true for large random graphs with high probability even though the conjectures were shown to fail for determinstic graphs. Next, we study asymptotic behaviour of some algebraic invariants—the Castelnuovo–Mumford regularity, projective dimension and depth—of such random edge ideals in the sparse regime (i.e., \(p = \frac{\lambda }{n}, \lambda \in (0,\infty )\) ). These invariants are studied using local weak convergence (or Benjamini-Schramm convergence) and relating them to invariants on Galton–Watson trees. We also show that when \(p \rightarrow 0\) or \(p \rightarrow 1\) fast enough, then with high probability the edge ideals are unmixed and for most other choices of p, these ideals are not unmixed with high probability. This is further progress towards the conjecture that random monomial ideals are unlikely to have Cohen–Macaulay property (De Loera et al. in Proc Am Math Soc 147(8):3239–3257, 2019; J Algebra 519:440–473, 2019) in the setting when the number of variables goes to infinity but the degree is fixed. PubDate: 2023-12-01
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Abstract: Abstract This paper presents new results on the identities satisfied by the sylvester and Baxter monoids. We show how to embed these monoids, of any rank strictly greater than 2, into a direct product of copies of the corresponding monoid of rank 2. This confirms that all monoids of the same family, of rank greater than or equal to 2, satisfy exactly the same identities. We then give a complete characterization of those identities, and prove that the varieties generated by the sylvester and the Baxter monoids have finite axiomatic rank, by giving a finite basis for them. PubDate: 2023-12-01
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Abstract: Abstract We introduce the concept of a rank-saturating system and outline its correspondence to a rank-metric code with a given covering radius. We consider the problem of finding the value of \(s_{q^m/q}(k,\rho )\) , which is the minimum \(\mathbb {F}_q\) -dimension of a q-system in \(\mathbb {F}_{q^m}^k\) that is rank- \(\rho \) -saturating. This is equivalent to the covering problem in the rank metric. We obtain upper and lower bounds on \(s_{q^m/q}(k,\rho )\) and evaluate it for certain values of k and \(\rho \) . We give constructions of rank- \(\rho \) -saturating systems suggested from geometry. PubDate: 2023-12-01
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Abstract: Abstract We prove various properties on the structure of groups whose power graph is chordal. Nilpotent groups with this property have been classified by (Electron J Combin 28(3):14, 2021). Here we classify the finite simple groups with chordal power graph, relative to typical number theoretic conditions. We do so by devising several sufficient conditions for the existence and non-existence of long cycles in power graphs of finite groups. We examine other natural group classes, including special linear, symmetric, generalized dihedral and quaternion groups, and we characterize direct products with chordal power graph. The classification problem is thereby reduced to directly indecomposable groups, and we further obtain a list of possible socles. Lastly, we give a general bound on the length of an induced path in chordal power graphs, providing another potential road to advance the classification beyond simple groups. PubDate: 2023-12-01
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Abstract: Abstract A connected graph \(\Gamma =(V,E)\) of valency at least 3 is called a basic 2-arc-transitive graph if its full automorphism group has a subgroup G with the following properties: (i) G acts transitively on the set of 2-arcs of \(\Gamma \) , and (ii) every minimal normal subgroup of G has at most two orbits on V. Based on Praeger’s theorems on 2-arc-transitive graphs, this paper presents a further understanding on the automorphism group of a basic 2-arc-transitive graph. PubDate: 2023-12-01
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Abstract: Abstract Let \(\mathcal{M}\) be an orientably regular (resp. regular) map with the number n vertices. By \(G^+\) (resp. G) we denote the group of all orientation-preserving automorphisms (resp. all automorphisms) of \(\mathcal{M}\) . Let \(\pi \) be the set of prime divisors of n. A Hall \(\pi \) -subgroup of \(G^+\) (resp. G) is meant a subgroup such that the prime divisors of its order all lie in \(\pi \) and the primes of its index all lie outside \(\pi \) . It is mainly proved in this paper that (1) suppose that \(\mathcal{M}\) is an orientably regular map where n is odd. Then \(G^+\) is solvable and contains a normal Hall \(\pi \) -subgroup; (2) suppose that \(\mathcal{M}\) is a regular map where n is odd. Then G is solvable if it has no composition factors isomorphic to \(\hbox {PSL}(2,q)\) for any odd prime power \(q\ne 3\) , and G contains a normal Hall \(\pi \) -subgroup if and only if it has a normal Hall subgroup of odd order. PubDate: 2023-11-23
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Abstract: Abstract Let V be a cubic surface defined by the equation \(T_0^3+T_1^3+T_2^3+\theta T_3^3=0\) over a quadratic extension of 3-adic numbers \(k=\mathbb {Q}_3(\theta )\) , where \(\theta ^3=1\) . We show that a relation on a set of geometric k-points on V modulo \((1-\theta )^3\) (in a ring of integers of k) defines an admissible relation and a commutative Moufang loop associated with classes of this admissible equivalence is non-associative. This answers a problem that was formulated by Yu. I. Manin more than 50 years ago about existence of a cubic surface with a non-associative Moufang loop of point classes. PubDate: 2023-11-08
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Abstract: Abstract Given a simple graph, consider the polynomial ring with coefficients in a field and variables identified with the edges of the graph. Given a non-empty even cardinality Eulerian subgraph and a choice of half of its edges, consider the homogeneous binomial obtained by taking the product of these edges minus the product of the remaining edges of the subgraph. We define a homogeneous ideal by taking as generators all binomials obtained in this way, varying the Eulerian subgraph and the choice of half of its edges, together with the squares of the variables of the ring. This ideal is related to the Eulerian ideal, introduced by Neves, Vaz Pinto and Villarreal. We call the corresponding quotient the Eulerian Artinian algebra associated with the graph. The goal of the present work is to study the socle of these algebras through the lens of graph theory. Our main results include a combinatorial characterization of a monomial basis of the socle, a characterization of Gorenstein Eulerian Artinian algebras in the case of bipartite graphs and the computation of the h-vector and socle degrees in the cases of a complete graph and a complete bipartite graph. PubDate: 2023-11-02
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Abstract: Abstract In this paper, we explicitly give combinatorial formulas for the regularity of powers of edge ideals, \({\text {reg}}(I(D)^k)\) , of weighted oriented unmixed forests D whose leaves are sinks ( \(V^+(D)\) are sinks). This combinatorial formula is a piecewise linear function of k, for \(k \ge 1\) . PubDate: 2023-11-01
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Abstract: Abstract Given a closed, convex cone \(K\subseteq \mathbb {R}^n\) , a multivariate polynomial \(f\in \mathbb {C}[\textbf{z}]\) is called K-stable if the imaginary parts of its roots are not contained in the relative interior of K. If K is the nonnegative orthant, K-stability specializes to the usual notion of stability of polynomials. We develop generalizations of preservation operations and of combinatorial criteria from usual stability toward conic stability. A particular focus is on the cone of positive semidefinite matrices (psd-stability). In particular, we prove the preservation of psd-stability under a natural generalization of the inversion operator. Moreover, we give conditions on the support of psd-stable polynomials and characterize the support of special families of psd-stable polynomials. PubDate: 2023-11-01
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Abstract: Abstract Sidon spaces have been introduced by Bachoc et al. (in: Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press, 2017) as the q-analogue of Sidon sets. The interest on Sidon spaces has increased quickly, especially after the work of Roth et al. (IEEE Trans Inform Theory 64(6):4412–4422, 2017), in which they highlighted the correspondence between Sidon spaces and cyclic subspace codes. Up to now, the known constructions of Sidon Spaces may be divided in three families: the ones contained in the sum of two multiplicative cosets of a fixed subfield of \(\mathbb {F}_{q^n}\) , the ones contained in the sum of more than two multiplicative cosets of a fixed subfield of \(\mathbb {F}_{q^n}\) and finally the ones obtained as the kernel of subspace polynomials. In this paper, we will mainly focus on the first class of examples, for which we provide characterization results and we will show some new examples, arising also from some well-known combinatorial objects. Moreover, we will give a quite natural definition of equivalence among Sidon spaces, which relies on the notion of equivalence of cyclic subspace codes and we will discuss about the equivalence of the known examples. PubDate: 2023-10-24
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Abstract: Abstract We show that the solutions to the equations, defining the so-called Calabi–Yau condition for fourth-order operators of degree two, define a variety that consists of ten irreducible components. These can be described completely in parametric form, but only two of the components seem to admit arithmetically interesting operators. We include a description of the 69 essentially distinct fourth-order Calabi–Yau operators of degree two that are presently known to us. PubDate: 2023-10-10
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Abstract: Abstract In the paper, we develop further the properties of Schur rings over infinite groups, with particular emphasis on the virtually cyclic group \(\mathcal {Z}\times \mathcal {Z}_p\) , where p is a prime. We provide structure theorems for primitive sets in these Schur rings. In the case of Fermat and safe primes, a complete classification theorem is proven, which states that all Schur rings over \(\mathcal {Z}\times \mathcal {Z}_p\) are traditional. We also draw analogs between Schur rings over \(\mathcal {Z}\times \mathcal {Z}_p\) and partitions of difference sets over \(\mathcal {Z}_p\) . PubDate: 2023-10-05
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Abstract: Abstract We extend Alon’s Combinatorial Nullstellensatz from polynomial rings over fields to polynomial rings over division rings and to rings of polynomial functions over centrally finite division algebras. We apply our results to extend classical theorems from additive number theory to the additive theory of division rings, where the size of algebraic sets is measured by their rank in the sense of Lam. PubDate: 2023-09-23
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Abstract: Abstract For a graph G with \(u,v\in V(G)\) , denote by \(d_G(u,v)\) the distance between u and v in G, which is the length of a shortest path connecting them if there is at least one path from u to v in G and is \(\infty \) otherwise. The closeness matrix of a graph G is the \( V(G) \times V(G) \) symmetric matrix \((c_G(u,v))_{u,v\in V(G}\) , where \(c_G(u,v)=2^{-d_G(u,v)}\) if \(u\ne v\) and 0 otherwise. The closeness eigenvalues of a graph G are the eigenvalues of C(G). We determine the graphs for which the second largest closeness eigenvalues belong to \(\left( -\infty , a\right) \) , where \(a\approx -0.1571\) is the second largest root of \(x^3-x^2-\frac{11}{8}x-\frac{3}{16}=0\) . We also identify the n-vertex graphs with a closeness eigenvalue of multiplicity \(n-1\) , \(n-2\) and \(n-3\) , respectively. PubDate: 2023-09-22
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