Journal of Algebraic Combinatorics
Journal Prestige (SJR): 1.199 Citation Impact (citeScore): 1 Number of Followers: 3 Hybrid journal (It can contain Open Access articles) ISSN (Print) 15729192  ISSN (Online) 09259899 Published by SpringerVerlag [2469 journals] 
 The lambda number of the power graph of a finite pgroup

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Abstract: Abstract An L(2, 1)labelling of a finite graph \(\varGamma \) is a function that assigns integer values to the vertices \(V(\varGamma )\) of \(\varGamma \) (colouring of \(V(\varGamma )\) by \({\mathbb {Z}}\) ) so that the absolute difference of two such values is at least 2 for adjacent vertices and is at least 1 for vertices, which are precisely distance 2 apart. The lambda number \(\lambda (\varGamma )\) of \(\varGamma \) measures the least number of integers needed for such a labelling (colouring). A power graph \(\varGamma _G\) of a finite group G is a graph with vertex set as the elements of G and two vertices are joined by an edge if and only if one of them is a positive integer power of the other. It is known that \(\lambda (\varGamma _G) \ge G \) for any finite group. In this paper, we show that if G is a finite group of a prime power order, then \(\lambda (\varGamma _G) = G \) if and only if G is neither cyclic nor a generalized quaternion 2group. This settles a partial classification of finite groups achieving the lower bound of lambda number.
PubDate: 20220919

 Cliff operads: a hierarchy of operads on words

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Abstract: Abstract A new hierarchy of operads over the linear spans of \(\delta \) cliffs, which are some words of integers, is introduced. These operads are intended to be analogues of the operad of permutations, also known as the associative symmetric operad. We obtain operads whose partial compositions can be described in terms of intervals of the lattice of \(\delta \) cliffs. These operads are very peculiar in the world of the combinatorial operads since, despite the relative simplicity for their construction, they are infinitely generated and they have nonquadratic and nonhomogeneous nontrivial relations. We provide a general construction for some of their quotients. We use it to endow the spaces of permutations, mincreasing trees, crectangular paths, and mDyck paths with operad structures. The operads on crectangular paths admit, as Koszul duals, operads generalizing the duplicial and triplicial operads.
PubDate: 20220918

 Typed angularly decorated planar rooted trees and generalized
Rota–Baxter algebras
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Abstract: Abstract We introduce a generalization of parametrized Rota–Baxter algebras, named \(\Omega \) Rota–Baxter algebra, which includes family and matching Rota–Baxter algebras. We study the structure needed on the set \(\Omega \) of parameters in order to obtain that free \(\Omega \) Rota–Baxter algebras are described in terms of typed and angularly decorated planar rooted trees: we obtain the notion of \(\lambda \) extended diassociative semigroup, which includes sets (for matching Rota–Baxter algebras) and semigroups (for family Rota–Baxter algebras), and many other examples. We also describe free commutative \(\Omega \) Rota–Baxter algebras generated by a commutative algebra A in terms of typed words.
PubDate: 20220914

 On Cayley representations of central Cayley graphs over almost simple
groups
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Abstract: Abstract A Cayley graph over a group G is said to be central if its connection set is a normal subset of G. We prove that every central Cayley graph over a simple group G has at most two pairwise nonequivalent Cayley representations over G associated with the subgroups of \({{\,\mathrm{Sym}\,}}(G)\) induced by left and right multiplications of G. We also provide an algorithm which, given a central Cayley graph \(\Gamma \) over an almost simple group G whose socle is of a bounded index, finds the full set of pairwise nonequivalent Cayley representations of \(\Gamma \) over G in time polynomial in size of G.
PubDate: 20220905

 Specializing Koornwinder polynomials to Macdonald polynomials of type
B, C, D and BC
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Abstract: Abstract We study the specializations of parameters in Koornwinder polynomials to obtain Macdonald polynomials associated to the subsystems of the affine root system of type \((C_n^\vee ,C_n)\) in the sense of Macdonald (Affine Hecke algebras and orthogonal polynomials, Cambridge tracts in mathematics, Cambridge Univ Press, 2003), and summarize them in what we call the specialization table. As a verification of our argument, we check the specializations to type B, C and D via Ram–Yip type formulas of nonsymmetric Koornwinder and Macdonald polynomials.
PubDate: 20220905

 The covering radius of permutation designs

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Abstract: Abstract A notion of tdesigns in the symmetric group on n letters was introduced by Godsil in 1988. In particular, ttransitive sets of permutations form a tdesign. We derive upper bounds on the covering radius of these designs, as a function of n and t and in terms of the largest zeros of Charlier polynomials.
PubDate: 20220903

 Fiber cones of rational normal scrolls are Cohen–Macaulay

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Abstract: Abstract In this paper, we show that the fiber cones of rational normal scrolls are Cohen–Macaulay. As an application, we compute their Castelnuovo–Mumford regularities and \(\varvec{a}\) invariants, as well as the reduction number of the defining ideals of the rational normal scrolls. We also characterize the Gorensteinness of the fiber cone.
PubDate: 20220901

 On maximal cliques of Cayley graphs over fields

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Abstract: Abstract We describe a new class of maximal cliques, with a vector space structure, of Cayley graphs defined on the additive group of a field. In particular, we show that in the cubic Paley graph with order \(q^3\) , the subfield with q elements forms a maximal clique. Similar statements also hold for quadruple Paley graphs and Peisert graphs with quartic order.
PubDate: 20220901

 Regular maps of 2power order

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Abstract: Abstract In this paper, we consider the possible types of regular maps of order \(2^n\) , where the order of a regular map is the order of automorphism group of the map. For \(n \le 11\) , M. Conder classified all regular maps of order \(2^n\) . It is easy to classify regular maps of order \(2^n\) whose valency or covalency is 2 or \(2^{n1}\) . So we assume that \(n \ge 12\) and \(2\le s,t\le n2\) with \(s\le t\) to consider regular maps of order \(2^n\) with type \(\{2^s, 2^t\}\) . We show that for \(s+t\le n\) or for \(s+t>n\) with \(s=t\) , there exists a regular map of order \(2^n\) with type \(\{2^s, 2^t\}\) , and furthermore, we classify regular maps of order \(2^n\) with types \(\{2^{n2},2^{n2}\}\) and \(\{2^{n3},2^{n3}\}\) . We conjecture that if \(s+t>n\) with \(s<t\) , then there is no regular map of order \(2^n\) with type \(\{2^s, 2^t\}\) , and we confirm the conjecture for \(t=n2\) and \(n3\) .
PubDate: 20220901

 A Littlewood–Richardson rule for Koornwinder polynomials

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Abstract: Abstract Koornwinder polynomials are qorthogonal polynomials equipped with extra five parameters and the \(B C_n\) type Weyl group symmetry, which were introduced by Koornwinder (Contemp Math 138:189–204, 1992) as multivariate analogue of Askey–Wilson polynomials. They are now understood as the Macdonald polynomials associated with the affine root system of type \((C^\vee _n,C_n)\) via the Macdonald–Cherednik theory of double affine Hecke algebras. In this paper, we give explicit formulas of Littlewood–Richardson coefficients for Koornwinder polynomials, i.e., the structure constants of the product as invariant polynomials. Our formulas are natural \((C^\vee _n,C_n)\) analogue of Yip’s alcovewalk formulas (Math Z 272:1259–1290, 2012) which were given in the case of reduced affine root systems.
PubDate: 20220901

 Equivalence of Butsontype Hadamard matrices

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Abstract: Abstract Two matrices \(H_1\) and \(H_2\) with entries from a multiplicative group G are said to be monomially equivalent, denoted by \(H_1\cong H_2\) , if one of the matrices can be obtained from the other via a sequence of row and column permutations and, respectively, left and rightmultiplication of rows and columns with elements from G. One may further define matrices to be Hadamard equivalent if \(H_1 \cong \phi (H_2)\) for some \(\phi \in \mathrm {Aut}(G)\) . For many classes of Hadamard and related matrices, it is straightforward to show that these are closed under Hadamard equivalence. It is here shown that also the set of Butsontype Hadamard matrices is closed under Hadamard equivalence.
PubDate: 20220901

 Freefermions and skew stable Grothendieck polynomials

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Abstract: Abstract We present a freefermionic presentation of the skew (dual) stable Grothendieck polynomials. A direct proof of their determinantal formulas is given from this presentation. We also introduce a combinatorial method to describe the multiplication map and its adjoint over the space of skew (dual) stable Grothendieck polynomials. This calculation requires the use of noncommutative supersymmetric Schur functions.
PubDate: 20220901

 On the Hopf algebra of multicomplexes

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Abstract: Abstract We introduce a general class of combinatorial objects, which we call multicomplexes, which simultaneously generalizes graphs, multigraphs, hypergraphs and simplicial and delta complexes. We introduce a natural algebra of multicomplexes which is defined as the algebra which has a formal basis \({\mathscr {C}}\) of all isomorphism types of multicomplexes, and multiplication is to take the disjoint union. This is a Hopf algebra with an operation encoding the disassembly information for such objects and extends the Hopf algebra of graphs. In our main result, we explicitly describe here the structure of this Hopf algebra of multicomplexes H. We find an explicit basis \({\mathscr {B}}\) of the space of primitives, which is of combinatorial relevance: it is such that each multicomplex is a polynomial with nonnegative integer coefficients of the elements of \({\mathscr {B}}\) , and each \(b\in {\mathscr {B}}\) is a polynomial with integer coefficients in \({\mathscr {C}}\) . Using this, we find the grouping free formula for the antipode. The coefficients appearing in all these polynomials are, up to signs, numbers counting multiplicities of submulticomplexes in a multicomplex. We also explicitly illustrate how our results specialize to the graph Hopf algebra, and observe how they specialize to results in all of the abovementioned particular cases. We also investigate applications of these results to the graph reconstruction conjectures and rederive some results in the literature on these questions.
PubDate: 20220901

 The module theory of semisymmetric quasigroups, totally symmetric
quasigroups, and triple systems
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Abstract: Abstract This paper initiates a unified module theory for four varieties of quasigroups: semisymmetric, semisymmetric idempotent, totally symmetric, and totally symmetric idempotent. These classes correspond, respectively, to extended Mendelsohn triple systems, Mendelsohn triple systems (MTS), extended Steiner triple systems, and Steiner triple systems (STS), which, in turn, correspond to partitions of complete (directed) graphs. Letting \({\mathbf {V}}\) stand for any of the aforementioned quasigroup categories, we determine a ring, \({\mathbb {Z}}{\mathbf {V}}\!{Q}\) , such that abelian groups in the slice category \({\mathbf {V}}\!/\!{Q}\) are equivalent to (right) \({\mathbb {Z}}{\mathbf {V}}\!{Q}\) modules. This ring is a quotient of the group algebra of the socalled universal stabilizer of the \({\mathbf {V}}\) quasigroup Q. We prove that the universal stabilizer of Q in \({\mathbf {V}}\) is the fundamental group of either the graph from which the triple system sources its blocks, or a closely related space. In each of the four varieties, we provide a free product description of \({\mathbb {Z}}{\mathbf {V}}\!{Q}\) , and show that the factorizations are indexed by blocks of the triple system. We also consider the ranks of free group rings that appear in the coproduct for \({\mathbb {Z}}{\mathbf {V}}\!{Q}\) (for the cases \({\mathbf {V}}=\mathbf {MTS}, \mathbf {STS}\) ), and we describe these ranks in terms of pentagonal numbers. We establish a relationship between the module theory of MTS and commutative Moufang loops of exponent 3. As an application, we show how the MTS module theory completely accounts for the distributive, nonmedial quasigroups of order 81.
PubDate: 20220901

 On joins of a clique and a coclique as star complements in regular graphs

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Abstract: Abstract In this paper we consider rregular graphs G that admit the vertex set partition such that one of the induced subgraphs is the join of an svertex clique and a tvertex coclique and represents a star complement for an eigenvalue \(\mu \) of G. The cases in which one of the parameters s, t is less than 2 or \(\mu =r\) are already resolved. It is conjectured in Wang et al. (Linear Algebra Appl 579:302–319, 2019) that if \(s, t\ge 2\) and \(\mu \ne r\) , then \(\mu =2, t=2\) and \(G=\overline{(s+1)K_2}\) . For \(\mu =t\) we verify this conjecture to be true. We further study the case in which \(\mu \ne t\) and confirm the conjecture provided \(t^24\mu ^2t4\mu ^3=0\) . For the remaining possibility we determine the structure of a putative counterexample and relate its existence to the existence of a particular 2class block design. It occurs that the smallest counterexample would have 1265 vertices.
PubDate: 20220901

 VertexFace/Zeta correspondence

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Abstract: Abstract We present the characteristic polynomial for the transition matrix of a vertexface walk on a graph, and obtain its spectra. Furthermore, we express the characteristic polynomial for the transition matrix of a vertexface walk on the 2dimensional torus by using its adjacency matrix, and obtain its spectra. As an application, we define a new walktype zeta function with respect to the transition matrix of a vertexface walk on the twodimensional torus, and present its explicit formula.
PubDate: 20220901

 Exceptional sequences of 8 line bundles on $$({\mathbb {P}}^1)^3$$ ( P 1 )
3
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Abstract: Abstract We investigate maximal exceptional sequences of line bundles on \(({\mathbb {P}}^1)^r\) , i.e., those consisting of \(2^r\) elements. For \(r=3\) we show that they are always full, meaning that they generate the derived category. Everything is done in the discrete setup: Exceptional sequences of line bundles appear as special finite subsets \({s}\) of the Picard group \({\mathbb {Z}}^r\) of \(({\mathbb {P}}^1)^r\) , and the question of generation is understood like a process of contamination of the whole \({\mathbb {Z}}^r\) out of an infectious seed \({s}\) .
PubDate: 20220901

 Comparing symbolic powers of edge ideals of weighted oriented graphs

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Abstract: Abstract Let D be a weighted oriented graph and I(D) be its edge ideal. If D contains an induced odd cycle of length \(2n+1\) , under certain condition, we show that \( {I(D)}^{(n+1)} \ne {I(D)}^{n+1}\) . We give necessary and sufficient condition for the equality of ordinary and symbolic powers of edge ideals of weighted oriented graphs having each edge in some induced odd cycle of it. We characterize the weighted naturally oriented unicyclic graphs with unique odd cycles and weighted naturally oriented even cycles for the equality of ordinary and symbolic powers of their edge ideals. Let \( D^{\prime } \) be the weighted oriented graph obtained from D after replacing the weights of vertices with nontrivial weights which are sinks, by trivial weights. We show that the symbolic powers of I(D) and \(I(D^{\prime })\) behave in a similar way. Finally, if D is any weighted oriented star graph, we prove that \( {I(D)}^{(s)} = {I(D)}^s \) for all \(s \ge 2.\)
PubDate: 20220901

 Maximal generating degrees of integral closures of powers of monomial
ideals
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Abstract: Abstract We give effective lower and upper bounds on the maximal generating degree \(d(\overline{I^n})\) and the Castelnuovo–Mumford regularity \({{\,\mathrm{reg}\,}}(\overline{I^n})\) of the integral closure of a power \(I^n\) of a monomial ideal I for all n and determine the leading coefficient of the asymptotically linear functions \(d(\overline{I^n})\) and \({{\,\mathrm{reg}\,}}(\overline{I^n})\) . A number \(n_0\) is also given such that \(d(\overline{I^n})\) becomes a linear function of n when \(n\ge n_0\) .
PubDate: 20220901

 Smallest graphs with given automorphism group

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Abstract: Abstract For a finite group G, denote by \(\alpha (G)\) the minimum number of vertices of any graph \(\Gamma \) having \(\mathrm {Aut}(\Gamma )\cong G\) . In this paper, we prove that \(\alpha (G)\le G \) , with specified exceptions. The exceptions include four infinite families of groups, and 17 other small groups. Additionally, we compute \(\alpha (G)\) for the groups G such that \(\alpha (G)> G \) where the value \(\alpha (G)\) was previously unknown.
PubDate: 20220901
