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- Cohen–Macaulay oriented graphs with large girth
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Abstract: Abstract We classify the Cohen–Macaulay weighted oriented graphs whose underlying graphs have girth at least 5.
PubDate: 2025-02-12
DOI: 10.1007/s10801-025-01385-8
- Residue formula for flag manifold of type A from wall-crossing
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Abstract: Abstract We consider equivariant integrals on flag manifolds of type A. Using a computational method inspired by the theory of wall-crossing formulas by Takuro Mochizuki, we re-prove residue formulas for equivariant integrals given by Weber and Zielenkiewicz. As an application, we give the determinantal formula of the Grothendieck polynomial by properly setting K theory classes.
PubDate: 2024-12-24
DOI: 10.1007/s10801-024-01378-z
- On matrix product factorization of graphs
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Abstract: Abstract In this paper, we explore the concept of the “matrix product of graphs,” initially introduced by Prasad, Sudhakara, Sujatha, and M. Vinay. This operation involves the multiplication of adjacency matrices of two graphs with assigned labels, resulting in a weighted digraph. Our primary focus is on identifying graphs that can be expressed as the graphical matrix product of two other graphs. Notably, we establish that the only complete graph fitting this framework is \(K_{4n+1}\) , and moreover, the factorization is not unique. In addition, the only complete bipartite graph that can be expressed as the graphical matrix product of two other graphs is \(K_{2n,2m}\) . Furthermore, we introduce several families of graphs that exhibit such factorization and, conversely, some families that do not admit any factorization such as wheel graphs, friendship graphs, and paths.
PubDate: 2024-12-02
DOI: 10.1007/s10801-024-01377-0
- The dimension of an orbitope based on a solution to the Legendre pair
problem-
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Abstract: Abstract The Legendre pair problem is a particular case of a rank-1 semidefinite description problem that seeks to find a pair of vectors \((\textbf{u},\textbf{v})\) each of length \(\ell \) such that the vector \((\textbf{u}^{\top },\textbf{v}^{\top })^{\top }\) satisfies the rank-1 semidefinite description. The group \((\mathbb {Z}_\ell \times \mathbb {Z}_\ell )\rtimes \mathbb {Z}^{\times }_\ell \) acts on the solutions satisfying the rank-1 semidefinite description by \( ((i,j),k)(\textbf{u},\textbf{v})=((i,k)\textbf{u},(j,k)\textbf{v}) \) for each \(((i,j),k) \in (\mathbb {Z}_\ell \times \mathbb {Z}_\ell )\rtimes \mathbb {Z}^{\times }_\ell \) . By applying the methods based on representation theory in Bulutoglu [Discrete Optim 45 (2022)], and results in Ingleton [J Lond Math Soc 1(4): 445–460, 1956] and Lam and Leung [J Algebra 224:91–109, 2000], for a given solution \((\textbf{u}^{\top },\textbf{v}^{\top })^{\top }\) satisfying the rank-1 semidefinite description, we show that the dimension of the convex hull of the orbit of \(\textbf{u}\) under the action of \(\mathbb {Z}_{\ell }\) or \(\mathbb {Z}_\ell \rtimes \mathbb {Z}^{\times }_\ell \) is \(\ell -1\) provided that \(\ell =p^n\) or \(\ell =pq^i\) for \(i=1,2\) , any positive integer n, and any two odd primes p, q. Our results lead to the conjecture that this dimension is \(\ell -1\) in both cases. We also show that the dimension of the convex hull of all feasible points of the Legendre pair problem of length \(\ell \) is \(2\ell -2\) provided that it has at least one feasible point.
PubDate: 2024-12-02
DOI: 10.1007/s10801-024-01367-2
- Smooth skew morphisms on semi-dihedral groups
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Abstract: Abstract A permutation \(\varphi \) on a group G is called a skew morphism of G if \(\varphi (1) = 1\) , and there exists an integer-valued function \(\pi : G \rightarrow Z_m\) , where m is the order of \(\varphi \) , such that \(\varphi (ab) = \varphi (a)\varphi ^{\pi (a)}(b)\) , for all \(a, b\in G\) . A skew morphism \(\varphi \) is smooth if the associated power function \(\pi \) of \(\varphi \) takes constant values on each orbit of \(\varphi \) . In this paper, we shall classify the smooth skew morphisms of semi-dihedral groups.
PubDate: 2024-12-01
DOI: 10.1007/s10801-024-01362-7
- Finite 4-geodesic-transitive graphs with bounded girth
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Abstract: Abstract Praeger and the first author in Jin and Praeger (J Combin Theory Ser A 178:105349, 2021) asked the following problem: classify s-geodesic-transitive graphs of girth \(2s-1\) or \(2s-2\) , where \(s=4,5,6,7,8\) . In this paper, we study the \(s=4\) case, that is, study the family of finite (G, 4)-geodesic-transitive graphs of girth 6 or 7 for some group G of automorphisms. A reduction result on this family of graphs is first given. Let N be a normal subgroup of G which has at least 3 orbits on the vertex set. We show that such a graph \(\Gamma \) is a cover of its quotient \(\Gamma _N\) modulo the N-orbits and either \(\Gamma _N\) is (G/N, s)-geodesic-transitive where \(s=\min \{4,\textrm{diam}(\Gamma _N)\}\ge 3\) , or \(\Gamma _N\) is a (G/N, 2)-arc-transitive strongly regular graph. Next, using the classification of 2-arc-transitive strongly regular graphs, we determine all the (G, 4)-geodesic-transitive covers \(\Gamma \) when \(\Gamma _N\) is strongly regular.
PubDate: 2024-12-01
DOI: 10.1007/s10801-024-01358-3
- On the classification of low-degree ovoids of $$Q^+(5,q)$$
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Abstract: Abstract Ovoids of the Klein quadric \(Q^+(5,q)\) of \(\textrm{PG}(5,q)\) have been studied in the last 40 years, also because of their connection with spreads of \(\textrm{PG}(3,q)\) and hence translation planes. Beside the classical example given by a three-dimensional elliptic quadric (corresponding to the regular spread of \(\textrm{PG}(3,q)\) ) many other classes of examples are known. First of all the other examples (beside the elliptic quadric) of ovoids of Q(4, q) give also examples of ovoids of \(Q^+(5,q)\) . To every ovoid of \(Q^+(5,q)\) two bivariate polynomials \(f_1(x,y)\) and \(f_2(x,y)\) can be associated. Another important class of ovoids of \(Q^+(5,q)\) is given by the ones associated to a flock of a three-dimensional quadratic cone and in this case \(f_1(x,y)=y+g(x)\) . In this paper, we classify such ovoids of \(Q^+(5,q)\) with the additional properties that \(\max \{\deg (f_1),\deg (f_2)\}<(\frac{1}{6.31}q)^{\frac{3}{13}}-1\) , that is \(f_1(x,y)\) and \(f_2(x,y)\) have “low degree" compared with q.
PubDate: 2024-12-01
DOI: 10.1007/s10801-024-01365-4
- From Pascal’s Theorem to the geometry of Ziegler’s line
arrangements-
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Abstract: Abstract Günter Ziegler has shown in 1989 that some homological invariants associated with the free resolutions of Jacobian ideals of line arrangements are not determined by combinatorics. His classical example involves hexagons inscribed in conics. Independently, Sergey Yuzvinsky has arrived in 1993 at the same type of line arrangements in order to show that formality is not determined by the combinatorics. In this note, we look into the geometry of such line arrangements and find out an unexpected relation to the classical Pascal’s theorem. Our results give information on the minimal degree of a Jacobian syzygy and on the formality of such hexagonal line arrangements in general, without an explicit choice for the six vertices of the hexagon.
PubDate: 2024-12-01
DOI: 10.1007/s10801-024-01360-9
- On the intersection spectrum of $${\text {PSL}}_2(q)$$
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Abstract: Abstract Given a group G and a subgroup \(H \le G\) , a set \(\mathcal {F}\subset G\) is called H-intersecting if for any \(g,g' \in \mathcal {F}\) , there exists \(xH \in G/H\) such that \(gxH=g'xH\) . The intersection density of the action of G on G/H by (left) multiplication is the rational number \(\rho (G,H)\) , equal to the maximum ratio \(\frac{ \mathcal {F} }{ H }\) , where \(\mathcal {F} \subset G\) runs through all H-intersecting sets of G. The intersection spectrum of the group G is then defined to be the set $$\begin{aligned} \sigma (G) := \left\{ \rho (G,H) : H\le G \right\} . \end{aligned}$$ It was shown by Bardestani and Mallahi-Karai (J Algebraic Combin, 42(1):111–128, 2015) that if \(\sigma (G) = \{1\}\) , then G is necessarily solvable. The natural question that arises is, therefore, which rational numbers larger than 1 belong to \(\sigma (G)\) , whenever G is non-solvable. In this paper, we study the intersection spectrum of the linear group \({\text {PSL}}_2(q)\) . It is shown that \(2 \in \sigma \left( {\text {PSL}}_2(q)\right) \) , for any prime power \(q\equiv 3 \pmod 4\) . Moreover, when \(q\equiv 1 \pmod 4\) , it is proved that \(\rho ({\text {PSL}}_2(q),H)=1\) , for any odd index subgroup H (containing \({\mathbb {F}}_q\) ) of the Borel subgroup (isomorphic to \({\mathbb {F}}_q\rtimes {\mathbb {Z}}_{\frac{q-1}{2}}\) ) consisting of all upper triangular matrices.
PubDate: 2024-12-01
DOI: 10.1007/s10801-024-01356-5
- On the subadditivity condition of edge ideal
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Abstract: Abstract Let \(S=K[x_1,\ldots ,x_n]\) , where K is a field, and \(t_i(S/I)\) denotes the maximal shift in the minimal graded free S-resolution of the graded algebra S/I at degree i, where I is an edge ideal. In this paper, we prove that if \(t_b(S/I)\ge \lceil \frac{3b}{2} \rceil \) for some \(b\ge 0\) , then the subadditivity condition \(t_{a+b}(S/I)\le t_a(S/I)+t_b(S/I)\) holds for all \(a\ge 0\) . In addition, we prove that \(t_{a+4}(S/I)\le t_a(S/I)+t_4(S/I)\) for all \(a\ge 0\) (the case \(b=0,1,2,3\) is known). We conclude that if the projective dimension of S/I is at most 9, then I satisfies the subadditivity condition.
PubDate: 2024-12-01
DOI: 10.1007/s10801-024-01363-6
- Canonical reduced words and signed descent length enumeration in Coxeter
groups-
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Abstract: Abstract Reifegerste and independently, Petersen and Tenner studied a statistic \({{\,\textrm{drops}\,}}\) on permutations in \(\mathfrak {S}_n\) . Two other studied statistics on \(\mathfrak {S}_n\) are \({{\,\textrm{depth}\,}}\) and \({{\,\textrm{exc}\,}}\) . Using descents in canonical reduced words of elements in \(\mathfrak {S}_n\) , we give an involution \(f_A: \mathfrak {S}_n \mapsto \mathfrak {S}_n\) that leads to a neat formula for the signed trivariate enumerator of \({{\,\textrm{drops}\,}}, {{\,\textrm{depth}\,}}, {{\,\textrm{exc}\,}}\) in \(\mathfrak {S}_n\) . This gives a simple formula for the signed univariate drops enumerator in \(\mathfrak {S}_n\) . For the type-B Coxeter group \(\mathfrak {B}_n\) , using similar techniques, we show analogous univariate results. For the type-D Coxeter group, we get analogous but inductive univariate results. Under the famous Foata-Zeilberger bijection \(\phi _{FZ}\) which takes permutations to restricted Laguerre histories, we show that permutations \(\pi \) and \(f_A(\pi )\) map to the same Motzkin path, but have different history components. Using the Foata-Zeilberger bijection, we also get a continued fraction for the generating function enumerating the pair of statistics \(\textrm{drops}\) and \(\textrm{MAD}\) . Graham and Diaconis determined the mean and the variance of the Spearman metric of disarray \(D(\pi )\) when one samples \(\pi \) from \(\mathfrak {S}_n\) at random. As an application of our results, we get the mean and variance of the statistic \(\textrm{drops}(\pi )\) when we sample \(\pi \) from \(\mathcal {A}_n\) at random.
PubDate: 2024-12-01
DOI: 10.1007/s10801-024-01359-2
- Invariants for incidence matrix of a tree
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Abstract: Abstract For an oriented tree, we compute several graph invariants, including the minimal norm of the generalized inverse and the norm of the Moore–Penrose inverse of its incidence matrix. We present equivalent characterizations of these invariants in terms of the minimal ratios of vector norms related to the oriented tree’s incidence matrix. We also provide a method to find the optimal vertex cut and edge cut with respect to these invariants.
PubDate: 2024-12-01
DOI: 10.1007/s10801-024-01361-8
- l-connectivity, l-edge-connectivity and spectral radius of graphs
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Abstract: Abstract Let G be a connected graph. The toughness of G is defined as \(t(G)=\min \left\{ \frac{ S }{c(G-S)}\right\} \) , in which the minimum is taken over all proper subsets \(S\subset V(G)\) such that \(c(G-S)\ge 2\) where \(c(G-S)\) denotes the number of components of \(G-S\) . Confirming a conjecture of Brouwer, Gu (SIAM J Discrete Math 35:948–952, 2021) proved a tight lower bound on toughness of regular graphs in terms of the second largest absolute eigenvalue. Fan, Lin and Lu (Eur J Combin 110:103701, 2023) then studied the toughness of simple graphs from the spectral radius perspective. While the toughness is an important concept in graph theory, it is also very interesting to study S for which \(c(G-S)\ge l\) for a given integer \(l\ge 2\) . This leads to the concept of the l-connectivity, which is defined to be the minimum number of vertices of G whose removal produces a disconnected graph with at least l components or a graph with fewer than l vertices. Gu (Eur J Combin 92:103255, 2021) discovered a lower bound on the l-connectivity of regular graphs via the second largest absolute eigenvalue. As a counterpart, we discover the connection between the l-connectivity of simple graphs and the spectral radius. We also study similar problems for digraphs and an edge version.
PubDate: 2024-12-01
DOI: 10.1007/s10801-024-01357-4
- A classification of regular maps with Euler characteristic a negative
prime cube-
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Abstract: Abstract For an odd prime \(p\ge 5\) , all regular maps \(\mathcal{M}\) on non-orientable surfaces with Euler characteristic \(-p^3\) are classified. Explicitly, it is proved that either \(\mathcal{M}\) has type \(\{4,m\}\) and \(\hbox {Aut}(\mathcal{M})\cong ({\mathbb {Z}}_2\times {\mathbb {Z}}_2)\rtimes {\mathbb {D}}_{2\,m}\) , where \(m\equiv 3(\hbox {mod }6)\) and \(m-4=p^3\) ; or \(\mathcal{M}\) has type \(\{2m,2n\}\) and \(\hbox {Aut}(\mathcal{M})\cong {\mathbb {D}}_{2\,m}\times {\mathbb {D}}_{2n}\) , where \(1<m<n\) , \(2\not \mid m\) , \(\gcd (m,n)=1\) and \(mn-m-n=p^3\) . In particular, there exists no such map provided \(p\equiv 1 (\hbox {mod }12)\) .
PubDate: 2024-12-01
DOI: 10.1007/s10801-024-01364-5
- A family of symmetric graphs in relation to 2-point-transitive linear
spaces-
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Abstract: Abstract A graph \(\Gamma \) is G-symmetric if it admits G as a group of automorphisms acting transitively on the set of arcs of \(\Gamma \) , where an arc is an ordered pair of adjacent vertices. Let \(\Gamma \) be a G-symmetric graph such that its vertex set admits a nontrivial G-invariant partition \(\mathcal {B}\) , and let \(\mathcal {D}(\Gamma , \mathcal {B})\) be the incidence structure with point set \(\mathcal {B}\) and blocks \(\{B\} \cup \Gamma _{\mathcal {B}}(\alpha )\) , for \(B \in \mathcal {B}\) and \(\alpha \in B\) , where \(\Gamma _{\mathcal {B}}(\alpha )\) is the set of blocks of \(\mathcal {B}\) containing at least one neighbor of \(\alpha \) in \(\Gamma \) . In this paper, we classify all G-symmetric graphs \(\Gamma \) such that \(\Gamma _{\mathcal {B}}(\alpha ) \ne \Gamma _{\mathcal {B}}(\beta )\) for distinct \(\alpha , \beta \in B\) , the quotient graph of \(\Gamma \) with respect to \(\mathcal {B}\) is a complete graph, and \(\mathcal {D}(\Gamma , \mathcal {B})\) is isomorphic to the complement of a (G, 2)-point-transitive linear space.
PubDate: 2024-11-29
DOI: 10.1007/s10801-024-01368-1
- Sharp volume and multiplicity bounds for Fano simplices
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Abstract: Abstract We present sharp upper bounds on the volume, Mahler volume and multiplicity for Fano simplices depending on the dimension and Gorenstein index. These bounds rely on the interplay between lattice simplices and unit fraction partitions. Moreover, we present an efficient procedure for explicitly classifying Fano simplices of any dimension and Gorenstein index, and we carry out the classification up to dimension four for various Gorenstein indices.
PubDate: 2024-11-29
DOI: 10.1007/s10801-024-01366-3
- The least eigenvalues of integral circulant graphs
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Abstract: Abstract The integral circulant graph \(\textrm{ICG}_n (D)\) has the vertex set \(Z_n = \{0, 1, 2, \ldots , n - 1\}\) , where vertices a and b are adjacent if \(\gcd (a-b,n)\in D\) , with \(D \subseteq \{d: d \mid n,\ 1\le d<n\}\) . In this paper, we establish that the minimal value of the least eigenvalues (minimal least eigenvalue) of integral circulant graphs \(\textrm{ICG}_n(D)\) , given an order n with its prime factorization \(p_1^{\alpha _1}\cdots p_k^{\alpha _k}\) , is equal to \(-\frac{n}{p_1}\) . Moreover, we show that the minimal least eigenvalue of connected integral circulant graphs \(\textrm{ICG}_n(D)\) of order n whose complements are also connected is equal to \(-\frac{n}{p_1}+p_1^{\alpha _1-1}\) . Finally, we determine the second minimal eigenvalue among all least eigenvalues within the class of connected integral circulant graphs of a prescribed order n and show it to be equal to \(-\frac{n}{p_1}+p_1-1\) or \(-\frac{n}{p_1}+1\) , depending on whether \(\alpha _1>1\) or not, respectively. In all the aforementioned tasks, we provide a complete characterization of graphs whose spectra contain these determined minimal least eigenvalues.
PubDate: 2024-11-29
DOI: 10.1007/s10801-024-01376-1
- The q, t-symmetry of the generalized q, t-Catalan number
$$C_{(k_1,k_2,k_3)}(q,t)$$-
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Abstract: Abstract We present two distinct proofs of the q, t-symmetry for the generalized q, t-Catalan number \(C_{\vec {k}}(q,t)\) , where \(\vec {k}=(k_1,k_2,k_3)\) . The first proof is derived through the application of MacMahon’s partition analysis. The second proof is established via a direct bijection.
PubDate: 2024-11-29
DOI: 10.1007/s10801-024-01374-3
- Terwilliger algebras constructed from Cayley tables of finite Bol loops
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Abstract: Abstract We describe the Terwilliger algebras of the Latin square association schemes arising from Cayley tables of Bol loops. These association schemes have four non-trivial classes. We give some necessary conditions involving Terwilliger algebras for a quasigroup to be a Bol loop.
PubDate: 2024-11-27
DOI: 10.1007/s10801-024-01370-7
- Quandle coloring quivers of general torus links by dihedral quandles
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Abstract: Abstract We completely characterize the coloring quivers of general torus links, T(p, q) where p is prime, by dihedral quandles by first exhausting all possible numbers of colorings, followed by determining the interconnections between colorings in each case. The quiver is obtained as a function of the number of colorings. The quiver always contains complete subgraphs, in particular, a complete subgraph corresponding to the trivial colorings, but the total number of subgraphs in the quiver and the weights of their edges vary depending on the number of colorings.
PubDate: 2024-11-26
DOI: 10.1007/s10801-024-01373-4
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