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Abstract: Abstract For a subgraph G of a complete graph \(K_n\) , the \(K_n\) -complement of G, denoted by \(K_n-G\) , is the graph obtained from \(K_n-G\) by removing all the edges of G. In this paper, we express the number of spanning trees of the \(K_n\) -complement \(K_n-G\) of a bipartite graph G in terms of the determinant of the biadjcency matrices of all induced balanced bipartite subgraphs of G, which are nonsingular, and we derive formulas of the number of spanning trees of \(K_n-G\) for various important classes of bipartite graphs G, some of which generalize some previous results. PubDate: 2024-06-01

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Abstract: Abstract Ariki and Mathas (Math Z 233(3):601–623, 2000) showed that the simple modules of the Ariki–Koike algebras \(\mathcal {H}_{\mathbb {C},v;Q_1,\ldots , Q_m}\big (G(m, 1, n)\big )\) (when the parameters are roots of unity and \(v \ne 1\) ) are labeled by the so-called Kleshchev multipartitions. This together with Ariki’s categorification theorem enabled Ariki and Mathas to obtain the generating function for the number of Kleshchev multipartitions by making use of the Weyl–Kac character formula. In this paper, we revisit their generating function relation for the \(v=-1\) case. In particular, this \(v=-1\) scenario is of special interest as the corresponding Kleshchev multipartitions are closely tied with generalized Rogers–Ramanujan type partitions when \(Q_1=\cdots =Q_a=-1\) and \(Q_{a+1}=\cdots =Q_m =1\) . Based on this connection, we provide an analytic proof of the result of Ariki and Mathas for the above choice of parameters. Our second objective is to investigate simple modules of the Ariki–Koike algebra in a fixed block, which are, as widely known, labeled by the Kleshchev multiparitions with a fixed partition residue statistic. This partition statistic is also studied in the works of Berkovich, Garvan, and Uncu. Employing their results, we provide two bivariate generating function identities for the \(m=2\) scenario. PubDate: 2024-05-31

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Abstract: Abstract Recently, H. Dao and R. Nair gave a combinatorial description of simplicial complexes \(\Delta \) such that the squarefree reduction of the Stanley–Reisner ideal of \(\Delta \) has the WLP in degree 1 and characteristic zero. In this paper, we apply the connections between analytic spread of equigenerated monomial ideals, mixed multiplicities and birational monomial maps to give a sufficient and necessary condition for the squarefree reduction \(A(\Delta )\) to satisfy the WLP in degree i and characteristic zero in terms of mixed multiplicities of monomial ideals that contain combinatorial information of \(\Delta \) , we call them incidence ideals. As a consequence, we give an upper bound to the possible failures of the WLP of \(A(\Delta )\) in degree i in positive characteristics in terms of mixed multiplicities. Moreover, we extend Dao and Nair’s criterion to arbitrary monomial ideals in positive odd characteristics. PubDate: 2024-05-21

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Abstract: Abstract In this paper, we study a distance-regular graph \(\Gamma \) having a nonintegral eigenvalue \(\theta \) and give some restriction for the intersection numbers. As an application, we show that (the collinearity graph of) a thick regular near hexagon has a nonintegral eigenvalue if and only if it is a generalized hexagon of order (s, 1) such that s is not a square. PubDate: 2024-05-21

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Abstract: Abstract A good range of problems on trees can be described by the following general setting: Given a bilinear map \(*:\mathbb {R}^d\times \mathbb {R}^d\rightarrow \mathbb {R}^d\) and a vector \(s\in \mathbb {R}^d\) , we need to estimate the largest possible absolute value g(n) of an entry over all vectors obtained from applying \(n-1\) applications of \(*\) to n instances of s. When the coefficients of \(*\) are nonnegative and the entries of s are positive, the value g(n) is known to follow a growth rate \(\lambda =\lim _{n\rightarrow \infty } \root n \of {g(n)}\) . In this article, we prove that for such \(*\) and s there exist nonnegative numbers \(r,r'\) and positive numbers \(a,a'\) so that for every n, $$\begin{aligned} a n^{-r}\lambda ^n\le g(n)\le a' n^{r'}\lambda ^n. \end{aligned}$$ While proving the upper bound, we actually also provide another approach in proving the limit \(\lambda \) itself. The lower bound is proved by showing a certain form of submultiplicativity for g(n). Corollaries include a lower bound and an upper bound for \(\lambda \) , which are followed by a good estimation of \(\lambda \) when we have the value of g(n) for an n large enough. PubDate: 2024-05-21

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Abstract: Abstract We propose versions of higher Bruhat orders for types B and C. This is based on a theory of higher Bruhat orders of type A and their geometric interpretations (due to Manin–Shekhtman, Voevodskii–Kapranov, and Ziegler), and on our study of the so-called symmetric cubillages of cyclic zonotopes. PubDate: 2024-05-17

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Abstract: Abstract In the wake of Dutta and Adhikari, who in 2020 used partial transposition in order to get pairs of cospectral graphs, we investigate partial transposition for Hermitian complex matrices. This allows us to construct infinite pairs of complex unit gain graphs (or \({\mathbb {T}}\) -gain graphs) sharing either the spectrum of the adjacency matrix or even the spectrum of all the generalized adjacency matrices. This investigation also sheds new light on the classical case, producing examples that were still missing even for graphs. Partial transposition requires a block structure of the matrix: we interpreted it as if coming from a composition of \({\mathbb {T}}\) -gain digraphs. By proposing a suitable definition of rigidity specifically for \({\mathbb {T}}\) -gain digraphs, we then produce the first examples of pairs of non-isomorphic graphs, signed graphs and \({\mathbb {T}}\) -gain graphs obtained via partial transposition of matrices whose blocks form families of commuting normal matrices. In some cases, the non-isomorphic graphs detected in this way turned out to be hardly distinguishable, since they share the adjacency, the Laplacian and the signless Laplacian spectrum, together with many non-spectral graph invariants. PubDate: 2024-05-16

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Abstract: Abstract We prove that every regular graph of valency at least four whose automorphism group contains a nilpotent subgroup acting transitively on the vertex set admits a nowhere-zero 3-flow. PubDate: 2024-05-16

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Abstract: Abstract The original motivation of this paper was to find the context-free grammar for the joint distribution of peaks and valleys on permutations. Although such attempt was unsuccessful, we can obtain noncommutative symmetric function identities for the joint distributions of several descent-based statistics, including peaks, valleys and even/odd descents, on permutations via Zhuang’s generalized run theorem. Our results extend in a unified way several generating function formulas exist in the literature, including formulas of Carlitz and Scoville (Discrete Math 5:45–59, 1973; J Reine Angew Math 265:110–137, 1974), J. Combin. Theory Ser. A, 20: 336-356 (1976), Zhuang (Adv Appl Math 90:86–144, 2017), Pan and Zeng (Adv Appl Math 104:85–99, 2019; Discrete Math 346:113575, 2023). As applications of these generating function formulas, Wachs’ involution and Foata–Strehl action on permutations, we also investigate the signed counting of even and odd descents, and of descents and peaks, which provide two generalizations of Désarménien and Foata’s classical signed Eulerian identity. PubDate: 2024-05-09

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Abstract: Abstract We continue the study of intersection bodies of polytopes, focusing on the behavior of IP under translations of P. We introduce an affine hyperplane arrangement and show that the polynomials describing the boundary of \(I(P+t)\) can be extended to polynomials in variables \(t\in \mathbb {R}^d\) within each region of the arrangement. In dimension 2, we give a full characterization of those polygons such that their intersection body is convex. We give a partial characterization for general dimensions. PubDate: 2024-05-09

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Abstract: Abstract In this paper we develop the theory of cyclic flats of q-matroids. We show that the cyclic flats, together with their ranks, uniquely determine a q-matroid and hence derive a new q-cryptomorphism. We introduce the notion of \(\mathbb {F}_{q^m}\) -independence of an \(\mathbb {F}_q\) -subspace of \(\mathbb {F}_q^n\) and we show that q-matroids generalize this concept, in the same way that matroids generalize the notion of linear independence of vectors over a given field. PubDate: 2024-05-09

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Abstract: Abstract The Möbius function of the subgroup lattice of a finite group has been introduced by Hall and applied to investigate several questions. In this paper, we consider the Möbius function defined on an order ideal related to the lattice of the subgroups of an irreducible subgroup G of the general linear group \(\textrm{GL}(n,q)\) acting on the n-dimensional vector space \(V=\mathbb {F}_q^n\) , where \(\mathbb {F}_q\) is the finite field with q elements. We find a relation between this function and the Euler characteristic of two simplicial complexes \(\Delta _1\) and \(\Delta _2\) , the former raising from the lattice of the subspaces of V, the latter from the subgroup lattice of G. PubDate: 2024-05-08

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Abstract: Abstract Suppose that \(\Delta \) is a thick, locally finite and locally s-arc transitive G-graph with \(s \ge 4\) . For a vertex z in \(\Delta \) , let \(G_z\) be the stabilizer of z and \(G_z^{[1]}\) the kernel of the action of \(G_z\) on the neighbours of z. We say \(\Delta \) is of pushing up type provided there exist a prime p and a 1-arc (x, y) such that \(C_{G_z}(O_p(G_z^{[1]})) \le O_p(G_z^{[1]})\) for \(z \in \{x,y\}\) and \(O_p(G_x^{[1]}) \le O_p(G_y^{[1]})\) . We show that if \(\Delta \) is of pushing up type, then \(O_p(G_x^{[1]})\) is elementary abelian and \(G_x/G_x^{[1]}\cong X\) with \( \textrm{PSL}_2(p^a)\le X \le \mathrm{P\Gamma L}_2(p^a)\) . PubDate: 2024-05-08

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Abstract: Abstract A Cayley digraph \(\textrm{Cay}(G,S)\) of a group G with respect to a subset S of G is called a CI-digraph if for every Cayley digraph \(\textrm{Cay}(G,T)\) isomorphic to \(\textrm{Cay}(G,S)\) , there exists an \(\alpha \in \textrm{Aut}(G)\) such that \(S^\alpha =T\) . For a positive integer m, G is said to have the m-DCI property if all Cayley digraphs of G with out-valency m are CI-digraphs. Li (European J Combin 18:655–665, 1997) gave a necessary condition for cyclic groups to have the m-DCI property, and in this paper, we find a necessary condition for dihedral groups to have the m-DCI property. Let \(\textrm{D}_{2n}\) be the dihedral group of order 2n, and assume that \(\textrm{D}_{2n}\) has the m-DCI property for some \(1 \le m\le n-1\) . It is shown that n is odd, and if further \(p+1\le m\le n-1\) for an odd prime divisor p of n, then \(p^2\not \mid n\) . Furthermore, if n is a power of a prime q, then \(\textrm{D}_{2n}\) has the m-DCI property if and only if either \(n=q\) , or q is odd and \(1\le m\le q\) . PubDate: 2024-05-08

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Abstract: Abstract The k-token graph \(F_k(G)\) of a graph G is the graph whose vertices are the k-subsets of vertices from G, two of which are adjacent whenever their symmetric difference is a pair of adjacent vertices in G. It was proved that the algebraic connectivity of \(F_k(G)\) equals the algebraic connectivity of G with a proof using random walks and interchange of processes on a weighted graph. However, no algebraic or combinatorial proof is known, and it would be a hit in the area. In this paper, we algebraically prove that the algebraic connectivity of \(F_k(G)\) equals the one of G for new infinite families of graphs, such as trees, some graphs with hanging trees, and graphs with minimum degree large enough. Some examples of these families are the following: the cocktail party graph, the complement graph of a cycle, and the complete multipartite graph. PubDate: 2024-05-07

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Abstract: Abstract The Terwilliger algebras of asymmetric association schemes of rank 3, whose nonidentity relations correspond to doubly regular tournaments, are shown to have thin irreducible modules, and to always be of dimension \(4k+9\) for some positive integer k. It is determined that asymmetric rank 3 association schemes of order up to 23 are determined up to combinatorial isomorphism by the list of their complex Terwilliger algebras at each vertex, but this is no longer true at order 27. To distinguish order 27 asymmetric rank 3 association schemes, it is shown using computer calculations that the list of rational Terwilliger algebras at each vertex will suffice. PubDate: 2024-05-07

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Abstract: Abstract The behavior of objects associated with general extended affine Lie algebras is typically distinct from their counterparts in affine Lie algebras. Our research focuses on studying characters and Cartan automorphisms, which appear in the study of Chevalley involutions and Chevalley bases for extended affine Lie algebras. We show that for almost all extended affine Lie algebras, any finite-order Cartan automorphism is diagonal, and its corresponding combinatorial map is a character. PubDate: 2024-05-06

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Abstract: Abstract We introduce the concept of pseudocover, which is a counterpart of cover, for symmetric graphs. The only known example of pseudocovers of symmetric graphs so far was given by Praeger, Zhou and the first-named author a decade ago, which seems technical and hard to extend to obtain more examples. In this paper, we present a criterion for a symmetric extender of a symmetric graph to be a pseudocover, and then apply it to produce various examples of pseudocovers, including (1) with a single exception, each Praeger–Xu’s graph is a pseudocover of a wreath graph; (2) each connected tetravalent symmetric graph with vertex stabilizer of size divisible by 32 has connected pseudocovers. PubDate: 2024-05-06

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Abstract: Abstract Given a finite abelian group G and cyclic subgroups A, B, C of G of the same order, we find necessary and sufficient conditions for A, B, C to admit a common transversal for the cosets they afford. For an arbitrary number of cyclic subgroups, we give a sufficient criterion when there exists a common complement. Moreover, in several cases where a common transversal exists, we provide concrete constructions. PubDate: 2024-05-01

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Abstract: Abstract We first determine the structure of the power digraphs of completely 0-simple semigroups, and then some properties of their power graphs are given. As the main result in this paper, using Cameron and Ghosh’s theorem about power graphs of abelian groups, we obtain a characterization that two \(G^{0}\) -normal completely 0-simple orthodox semigroups S and T with abelian group \(\mathcal {H}\) -classes are isomorphic based on their power graphs. We also present an algorithm to determine that S and T are isomorphic or not. PubDate: 2024-05-01