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Abstract: Abstract The original version of this article, published online on 25 October 2021 (https://doi.org/10.1007/s10801-021-01054-6), has some imprecisions that are described and fixed in this document. PubDate: 2022-08-01

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Abstract: Abstract Double-threshold graphs are defined in terms of two real thresholds that break the real line into three regions, alternating as NO-YES-NO. If real ranks can be assigned to the vertices of a graph in such a way that two vertices are adjacent iff the sum of their ranks lies in the YES region, then that graph is a double-threshold graph with respect to the given set of thresholds. We demonstrate that all double-threshold graphs are permutation graphs. As a partial converse, we show that every bipartite permutation graph has a balanced double-threshold representation. That is, the vertices with negative rank form one part of the bipartition, those with positive rank the other part. PubDate: 2022-08-01

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Abstract: Abstract Let p be a prime. Cubic edge-transitive graphs of order \(2^np\) have been classified for \(1\le n\le 4\) in the literature. This paper is devoted to a general study of cubic edge-transitive graphs of order \(2^np\) . We first show that every connected cubic edge-transitive graph of order \(2^np\) with \(p>7\) is an edge-transitive N-cover of a connected cubic symmetric graph \(\Lambda _{2p}\) of order 2p, where N is a 2-group such that \(N/\Phi (N)\cong {{\mathbb {Z}}}_2^d\) with either \(d\le 2\) or \(d\ge 9\) . For the case when \(d\le 2\) , we give a characterization of edge-transitive N-covers of \(\Lambda _{2p}\) , and for the case when \(d\ge 9\) , we give a classification of edge-transitive N-covers of \(\Lambda _{2p}\) when p is a Zsigmondy prime of \( N -1\) . PubDate: 2022-08-01

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Abstract: Abstract A pointed graph \((\Gamma ,v_0)\) induces a family of transition matrices in Wildberger’s construction of a hermitian hypergroup using a random walk on \(\Gamma \) starting from \(v_0\) . In this study, we propose a necessary condition for producing a hermitian hypergroup considering a weaker condition than the distance-regularity for \((\Gamma ,v_0)\) . Furthermore, we show that the condition obtained connects the transition and adjacency matrices associated with \(\Gamma \) . PubDate: 2022-08-01

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Abstract: Tropical ideals are a class of ideals in the tropical polynomial semiring that combinatorially abstracts the possible collections of supports of all polynomials in an ideal over a field. We study zero-dimensional tropical ideals I with Boolean coefficients in which all underlying matroids are paving matroids, or equivalently, in which all polynomials of minimal support have support of size \(\deg (I)\) or \(\deg (I)+1\) —we call them paving tropical ideals. We show that paving tropical ideals of degree \(d+1\) are in bijection with \({\mathbb {Z}}^n\) -invariant d-partitions of \({\mathbb {Z}}^n\) . This implies that zero-dimensional tropical ideals of degree 3 with Boolean coefficients are in bijection with \({\mathbb {Z}}^n\) -invariant 2-partitions of quotient groups of the form \({\mathbb {Z}}^n/L\) . We provide several applications of these techniques, including a construction of uncountably many zero-dimensional degree-3 tropical ideals in one variable with Boolean coefficients, and new examples of non-realizable zero-dimensional tropical ideals. PubDate: 2022-08-01

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Abstract: Abstract We prove the alternating sign conjecture for the perfect matching derangement graph. PubDate: 2022-08-01

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Abstract: Abstract Hecke symmetries give rise to a family of graded algebras which represent quantum groups and spaces of noncommutative geometry. The present paper continues the work aiming to understand general properties of these algebras without a restriction on the parameter q of Hecke relation used in earlier results. However, if q is a root of 1, we need a restriction on the indecomposable modules for the Hecke algebras of type A that can occur as direct summands of representations in the tensor powers of the initial vector space V. In this setting, we generalize known results on rationality of Hilbert series. The combinatorial nature of this problem stems from a relationship between the Grothendieck ring of the category of comodules for the Faddeev–Reshetikhin–Takhtajan bialgebra A(R) associated with a Hecke symmetry R and the ring of symmetric functions. We then improve two results on monoidal equivalences of corepresentation categories and on Gorensteinness of graded algebras from a previous article. PubDate: 2022-08-01

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Abstract: A set \(S\subseteq V\) is independent in a graph \(G=\left( V,E\right) \) if no two vertices from S are adjacent. The independence number \(\alpha (G)\) is the cardinality of a maximum independent set, while \(\mu (G)\) is the size of a maximum matching in G. If \(\alpha (G)+\mu (G)\) equals the order of G, then G is called a König–Egerváry graph (Deming in Discrete Math 27:23–33, 1979; Sterboul in J Combin Theory Ser B 27:228–229, 1979). The number \(d\left( G\right) =\max \{\left A\right -\left N\left( A\right) \right :A\subseteq V\}\) is called the critical difference of G (Zhang in SIAM J Discrete Math 3:431–438, 1990) (where \(N\left( A\right) =\left\{ v:v\in V,N\left( v\right) \cap A\ne \emptyset \right\} \) ). It is known that \(\alpha (G)-\mu (G)\le d\left( G\right) \) holds for every graph (Levit and Mandrescu in SIAM J Discrete Math 26:399–403, 2012; Lorentzen in Notes on covering of arcs by nodes in an undirected graph, Technical report ORC 66-16. University of California, Berkeley, CA, Operations Research Center, 1966; Schrijver in Combinatorial optimization. Springer, Berlin, 2003). In Levit and Mandrescu (Graphs Combin 28:243–250, 2012), it was shown that \(d(G)=\alpha (G)-\mu (G)\) is true for every König–Egerváry graph. A graph G is (i) unicyclic if it has a unique cycle and (ii) almost bipartite if it has only one odd cycle. It was conjectured in Levit and Mandrescu (in: s of the SIAM conference on discrete mathematics, Halifax, Canada, p 40, abstract MS21, 2012, 3rd international conference on discrete mathematics, June 10–14, Karnatak University. Dharwad, India, 2013) and validated in Bhattacharya et al. (Discrete Math 341:1561–1572, 2018) that \(d(G)=\alpha (G)-\mu (G)\) holds for every unicyclic non-König–Egerváry graph G. In this paper, we prove that if G is an almost bipartite graph of order \(n\left( G\right) \) , then \(\alpha (G)+\mu (G)\in \left\{ n\left( G\right) -1,n\left( G\right) \right\} \) . Moreover, for each of these two values, we characterize the corresponding graphs. Further, using these findings, we show that the critical difference of an almost bipartite graph G satisfies $$\begin{aligned} d(G)=\alpha (G)-\mu (G)=\left \mathrm {core}(G)\right -\left N(\mathrm {core}(G))\right , \end{aligned}$$ where by core \(\left( G\right) \) we mean the intersection of all maximum independent sets. PubDate: 2022-08-01

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Abstract: Abstract Let \((A,+)\) be a finite Abelian group. Take the elements of A to be vertices of a complete graph and color the edge ab with \(a+b\) . A tree in A is rainbow colored provided all of its edges have different colors. In this paper, we study conditions that regulate whether or not a given tree can be realized as a rainbow spanning subtree of an Abelian group of the same order. For example, let \(C[h_1, \dots , h_s]\) denote the caterpillar with s spine vertices and with \(h_i\) hairs on the ith spine vertex. We characterize, by means of divisibility conditions, when a caterpillar of type \(C[k,\ell ], C[k,0,\ell ]\) or of type \(C[k,0,0,\ell ]\) embeds as a rainbow spanning tree in a group of the same order. We also show that embeddability as a rainbow spanning tree is not a local condition. That is, given any tree T and sufficiently large non-cyclic group A, some trees of order \(\left A \right \) that contain T as a subtree do embed as rainbow spanning trees in A, and some do not. For non-Boolean groups A of order at most 20, we give a complete catalogue of all trees that fail to embed as rainbow spanning trees of A. We also show that all rainbow spanning trees in A can be obtained from the star with center 0 through a simple pivoting procedure. PubDate: 2022-08-01

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Abstract: Abstract Brenti conjectured that, for any finite Coxeter group, the descent generating polynomial has only real zeros, and he left the type D case open. Dilks, Petersen, and Stembridge proposed a companion conjecture, which states that, for any irreducible finite Weyl group, the affine descent generating polynomial has only real zeros, and they left the type B and type D cases open. By developing the theory of \({\mathbf {s}}\) -Eulerian polynomials, Savage and Visontai confirmed the type D case of the former conjecture and the type B case of the latter conjecture. In this paper, based on the Hermite–Biehler theorem and the theory of linear transformations preserving Hurwitz stability, we obtain the Hurwitz stability of certain polynomials related to the descent generating polynomials of type D, and thus give an alternative proof of Savage and Visontai’s results. This new approach also enables us to prove Hyatt’s conjectures on the interlacing property of half Eulerian polynomials of type B and type D, and to prove that the h-polynomial of certain subcomplexes of Coxeter complexes of type D has only real zeros. As one of the most important application of this approach, we completely confirm Dilks, Petersen, and Stembridge’s conjecture by proving the real-rootedness of the affine descent generating polynomials of type D. PubDate: 2022-08-01

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Abstract: Abstract We determine the number of cubic surfaces with 27 lines over a finite field \({{\mathbb {F}}}_q\) . This is based on exploiting the relationship between non-conical six-arcs in a projective plane embedded in projective three-space and cubic surfaces with 27 lines. We revisit this classical relationship, which goes back to work of Clebsch in the nineteenth century. Our result can be used as an enumerative check for a computer classification of cubic surfaces with 27 lines over finite fields. PubDate: 2022-08-01

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Abstract: Abstract By appropriately choosing a defining set, we define a class of linear codes and establish their complete weight enumerators and weight enumerators using Weil sums. They only have three nonzero weights. This paper generalizes some results in Jian et al. (Finite Fields Appl 57:92–107, 2019). PubDate: 2022-08-01

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Abstract: Abstract Let \(\frac{1}{2}{\overline{H}}(2n,2)\) denote the halved folded 2n-cube with vertex set X and let \(T{:}{=}T(x)\) denote the Terwilliger algebra of \(\frac{1}{2}{\overline{H}}(2n,2)\) with respect to a fixed vertex x. In this paper, we assume \(n\ge 4\) and show that T coincides with the centralizer algebra of the stabilizer of x in the automorphism group of \(\frac{1}{2}{\overline{H}}(2n,2)\) by considering the action of this automorphism group on the set \(X\times X\times X\) . Then, we further describe the structure of T for the case \(n=2D\) and \(D\ge 3\) . The decomposition of T will be given by using the homogeneous components of \(V{:}{=}{\mathbb {C}}^X\) , each of which is a nonzero subspace of V spanned by the irreducible T-modules that are isomorphic. Moreover, we display a computable basis for every homogeneous component of V. PubDate: 2022-08-01

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Abstract: Abstract A skew morphism of a finite group A is a permutation \(\varphi \) on A fixing the identity element of A, and for which there exists an integer-valued function \(\pi :A\rightarrow {\mathbb {Z}}_{ \varphi }\) on A such that \(\varphi (ab)=\varphi (a)\varphi ^{\pi (a)}(b)\) for all \(a,b\in A\) . Moreover, the period of \(\varphi \) is the smallest positive integer d such that \(\pi (\varphi ^d(a))\equiv \pi (a)\pmod { \varphi }\) for all \(a\in A\) . In the case where \(d=1\) , the skew morphism \(\varphi \) is called smooth. It is well known that if \(\varphi \) is a skew morphism of period d, then \(\varphi ^d\) is a smooth skew morphism. Thus, every skew morphism of period d may be extracted as a dth root of a smooth skew morphism. In this paper, we introduce a new concept of average function to investigate skew morphisms and as an application we present a classification of smooth skew morphisms of the dicyclic groups. PubDate: 2022-07-29

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Abstract: Abstract We investigate what information on the orbit type stratification of a torus action on a compact space is contained in its rational equivariant cohomology algebra. Regarding the (labelled) poset structure of the stratification, we show that equivariant cohomology encodes the subposet of ramified elements. For equivariantly formal actions, we also examine what cohomological information of the stratification is encoded. In the smooth setting, we show that under certain conditions—which in particular hold for a compact orientable manifold with discrete fixed point set—the equivariant cohomologies of the strata are encoded in the equivariant cohomology of the manifold. PubDate: 2022-07-25

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Abstract: Abstract We prove that every Riordan array over \(\mathbb {C}\) whose main diagonal consists only of ones can be written as a product of at most five Riordan arrays of finite orders. PubDate: 2022-07-11

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Abstract: We consider a q-analogue of abstract simplicial complexes, called q-complexes, and discuss the notion of shellability for such complexes. It is shown that q-complexes formed by independent subspaces of a q-matroid are shellable. Further, we explicitly determine the homology of q-complexes corresponding to uniform q-matroids. We also outline some partial results concerning the determination of homology of arbitrary shellable q-complexes. PubDate: 2022-07-09

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Abstract: Abstract We study defining inequalities of string cones via a potential function on a reduced double Bruhat cell. We give a necessary criterion for the potential function to provide a minimal set of inequalities via tropicalization and conjecture an equivalence. PubDate: 2022-07-06

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