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Abstract: Abstract Quite a lot of attention has been paid recently to the classification of symmetric Cayley graphs of non-abelian simple groups. Besides the known complete classification on the cubic case, in most cases, the classifications are conditional with restrictions, such as on specified non-abelian simple groups or on solvable vertex-stabilizers. In this paper, a characterization of the 7-valent symmetric Cayley graphs of finite simple groups is given. PubDate: 2022-06-20
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Abstract: Abstract We show that the \(\mathrm {v}\) -number of an arbitrary monomial ideal is bounded below by the \(\mathrm {v}\) -number of its polarization and also find a criteria for the equality. By showing the additivity of associated primes of monomial ideals, we obtain the additivity of the v-numbers for arbitrary monomial ideals. We prove that the \(\mathrm {v}\) -number \(\mathrm {v}(I(G))\) of the edge ideal I(G), the induced matching number \(\mathrm {im}(G)\) and the regularity \(\mathrm {reg}(R/I(G))\) of a graph G, satisfy \(\mathrm {v}(I(G))\le \mathrm {im}(G)\le \mathrm {reg}(R/I(G))\) , where G is either a bipartite graph, or a \((C_{4},C_{5})\) -free vertex decomposable graph, or a whisker graph. There is an open problem in Jaramillo and Villarreal (J Combin Theory Ser A 177:105310, 2021), whether \(\mathrm {v}(I)\le \mathrm {reg}(R/I)+1\) , for any square-free monomial ideal I. We show that \(\mathrm {v}(I(G))>\mathrm {reg}(R/I(G))+1\) , for a disconnected graph G. We derive some inequalities of \(\mathrm {v}\) -numbers which may be helpful to answer the above problem for the case of connected graphs. We connect \(\mathrm {v}(I(G))\) with an invariant of the line graph L(G) of G. For a simple connected graph G, we show that \(\mathrm {reg}(R/I(G))\) can be arbitrarily larger than \(\mathrm {v}(I(G))\) . Also, we try to see how the \(\mathrm {v}\) -number is related to the Cohen–Macaulay property of square-free monomial ideals. PubDate: 2022-06-20
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Abstract: Abstract A 2-cell embedding of a graph into a nonorientable closed surface is called regular if its automorphism group acts regularly on its flags (incident vertex-edge-face triples). This paper characterizes automorphism group G of the nonorientable regular embeddings of simple graphs of order \(p^3\) , where p is a prime. PubDate: 2022-06-01
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Abstract: Abstract Let I be a square-free monomial ideal of projective dimension one. Starting with the Taylor complex on the generators of \(I^r\) , we use discrete Morse theory to describe a CW complex that supports a minimal free resolution of \(I^r\) . To do so, we concretely describe the acyclic matching on the faces of the Taylor complex. PubDate: 2022-06-01
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Abstract: Abstract A finite simple graph is called a k-multicirculant if its automorphism group contains a cyclic semiregular subgroup having k orbits on the vertex set. It was shown by Giudici et al. that, if k is squarefree and coprime to 6, then a cubic arc-transitive k-multicirculant has at most \(6k^2\) vertices (J. Combin. Theory Ser. B, 2017). In this paper, we classify the latter graphs under the assumption that their semiregular cyclic subgroups are contained in a soluble group of automorphisms acting transitively on the arc set of the graphs. As an application, cubic arc-transitive p-multicirculants are completely described for each odd prime p. PubDate: 2022-06-01
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Abstract: Abstract We introduce the concept of alternate-edge-colourings for maps and study highly symmetric examples of such maps. Edge-biregular maps of type (k, l) occur as smooth normal quotients of a particular index two subgroup of \(T_{k,l}\) , the full triangle group describing regular plane (k, l)-tessellations. The resulting colour-preserving automorphism groups can be generated by four involutions. We explore special cases when the usual four generators are not distinct involutions, with constructions relating these maps to fully regular maps. We classify edge-biregular maps when the supporting surface has non-negative Euler characteristic, and edge-biregular maps on arbitrary surfaces when the colour-preserving automorphism group is isomorphic to a dihedral group. PubDate: 2022-06-01
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Abstract: Abstract The cut sets of a graph are special sets of vertices whose removal disconnects the graph. They are fundamental in the study of binomial edge ideals, since they encode their minimal primary decomposition. We introduce the class of accessible graphs as the graphs with unmixed binomial edge ideal and whose cut sets form an accessible set system. We prove that the graphs whose binomial edge ideal is Cohen–Macaulay are accessible and we conjecture that the converse holds. We settle the conjecture for large classes of graphs, including chordal and traceable graphs, providing a purely combinatorial description of Cohen–Macaulayness. The key idea in the proof is to show that both properties are equivalent to a further combinatorial condition, which we call strong unmixedness. PubDate: 2022-06-01
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Abstract: Abstract Hoffman and Smith proved that in a graph with maximum degree \(\Delta \) if all edges are subdivided infinitely many times, then the largest eigenvalue, also called index, of the adjacency matrix converges to \(\Delta /(\sqrt{\Delta -1})\) . For the (signless) Laplacian of graphs, a similar result holds and the limit value is its square \(\Delta ^2/(\Delta -1)\) . Throughout the years, several scholars have progressed into characterizing the (connected) graphs whose adjacency or (signless) Laplacian index does not exceed the Hoffman–Smith limit value for \(\Delta =3\) , still there is not a complete characterization of such graphs. Here, we consider the signless Laplacian variant of this problem, and we characterize a large portion of such graphs. Also, we provide a structural restriction for the graphs not yet included for the complete characterization. Finally, we discuss the consequences on the adjacency variant of this problem. PubDate: 2022-06-01
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Abstract: Abstract Recently, Li and Pott proposed a new concept of intersection distribution concerning the interaction between the graph \(\{(x,f(x))~ ~x\in {\mathbb {F}}_{q}\}\) of f and the lines in the classical affine plane AG(2, q). Later, Kyureghyan et al. proceeded to consider the next simplest case, and derived the intersection distribution for all degree three polynomials over \({\mathbb {F}}_{q}\) with q both odd and even. They also proposed several conjectures therein. In this paper, we completely solve two conjectures of Kyureghyan et al. Namely, we prove two classes of power functions having intersection distribution: \(v_{0}(f)=\frac{q(q-1)}{3},~v_{1}(f)=\frac{q(q+1)}{2},~v_{2}(f)=0,~v_{3}(f)=\frac{q(q-1)}{6}\) . We mainly make use of the multivariate method and a certain type of equivalence on 2-to-1 mappings. The key point of our proof is to consider the number of the solutions of some low-degree equations. PubDate: 2022-06-01
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Abstract: Abstract A permutation \(\pi \) of a multiset is said to be a quasi-Stirling permutation if there do not exist four indices \(i<j<k<\ell \) such that \(\pi _i=\pi _k\) , \(\pi _j=\pi _{\ell }\) and \(\pi _i\ne \pi _j\) . Define $$\begin{aligned} \overline{Q}_{\mathcal {M}}(t,u,v)=\sum _{\pi \in \overline{\mathcal {Q}}_{\mathcal {M}}}t^{\mathrm{des}(\pi )}u^{\mathrm{asc}(\pi )}v^{\mathrm{plat}(\pi )}, \end{aligned}$$ where \(\overline{\mathcal {Q}}_{\mathcal {M}}\) denotes the set of quasi-Stirling permutations of the multiset \(\mathcal {M}\) , and \(\mathrm{asc}(\pi )\) (resp. \(\mathrm{des}(\pi )\) , \(\mathrm{plat}(\pi )\) ) denotes the number of ascents (resp. descents, plateaux) of \(\pi \) . Denote by \(\mathcal {M}^{\sigma }\) the multiset \(\{1^{\sigma _1}, 2^{\sigma _2}, \ldots , n^{\sigma _n}\}\) , where \(\sigma =(\sigma _1, \sigma _2, \ldots , \sigma _n)\) is an n-composition of K for positive integers K and n. In this paper, we show that \(\overline{Q}_{\mathcal {M}^{\sigma }}(t,u,v)=\overline{Q}_{\mathcal {M}^{\tau }}(t,u,v)\) for any two n-compositions \(\sigma \) and \(\tau \) of K. This is accomplished by establishing an \((\mathrm{asc}, \mathrm{des}, \mathrm{plat})\) -preserving bijection between \(\overline{\mathcal {Q}}_{\mathcal {M}^{\sigma }}\) and \(\overline{\mathcal {Q}}_{\mathcal {M}^{\tau }}\) . As applications, we obtain generalizations of several results for quasi-Stirling permutations on \(\mathcal {M}=\{1^k,2^k, \ldots , n^k\}\) obtained by Elizalde and solve an open problem posed by Elizalde. PubDate: 2022-06-01
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Abstract: Abstract By means of a new notion of subforests of an angularly decorated rooted forest, we give a combinatorial construction of a coproduct on the free Rota–Baxter algebra on angularly decorated rooted forests. We show that this coproduct equips the Rota–Baxter algebra with a bialgebra structure and further a Hopf algebra structure. PubDate: 2022-06-01
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Abstract: Abstract In this article, a possible supercharacter theory based on the degrees of the irreducible characters of a finite group is introduced. An investigation is then included into which finite groups allow this particular supercharacter theory. In particular, it is shown that groups with at most seven conjugacy classes must allow this supercharacter theory. However, there are groups with eight conjugacy classes that do not have this supercharacter theory. In addition, it is shown that semidirect products \(A \rtimes B\) where both A and B are cyclic groups and at least one is of prime order always allow this type of supercharacter theory. PubDate: 2022-06-01
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Abstract: Abstract In this note, we study factors of some alternating sums of products of q-binomial coefficients related to q-Narayana numbers. Let \({n\brack k}\) denote the q-binomial coefficients. We prove that for all positive integers \(n_1, \ldots , n_m\) , \(n_{m+1}=n_1\) , and \(j=0\) or \(2m-1\) , the alternating sum $$\begin{aligned} {n_1+n_m+1\brack n_1}^{-1}\sum _{k=-n_1}^{n_1}(-1)^k q^{jk^2+{k\atopwithdelims ()2}} \prod _{i=1}^m {n_i+n_{i+1}+1\brack n_i+k}{n_i+n_{i+1}+1\brack n_i+k+1} \end{aligned}$$ is a polynomial in q with integer coefficients, and it has non-negative coefficients if m is odd. This partially confirms a conjecture of Guo and Jiang. PubDate: 2022-06-01
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Abstract: Abstract Let G be a finite group. Let \(\pi \) be a permutation from \(S_{n}\) . We study the distribution of probabilities of equality \( a_{1}a_{2}\cdots a_{n-1}a_{n}=a_{\pi _{1}}^{\epsilon _{1} }a_{\pi _{2}}^{\epsilon _{2}}\cdots a_{\pi _{n-1}}^{\epsilon _{n-1}}a_{\pi _{n} }^{\epsilon _{n}},\) when \(\pi \) varies over all the permutations in \(S_{n}\) , and \(\epsilon _{i}\) varies over the set \(\{+1, -1\}\) . By [7], the case where all \(\epsilon _{i}\) are \(+1\) led to a close connection to Hultman numbers. In this paper, we generalize the results, permitting \(\epsilon _{i}\) to be \(-1\) . We describe the spectrum of the probabilities of signed permutation equalities in a finite group G. This spectrum turns out to be closely related to the partition of \(2^{n}\cdot n!\) into a sum of the corresponding signed Hultman numbers. PubDate: 2022-06-01
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Abstract: Abstract George Andrews and Ae Ja Yee recently established beautiful results involving bivariate generalizations of the third-order mock theta functions \(\omega (q)\) and \(\nu (q)\) , thereby extending their earlier results with the second author. Generalizing the Andrews–Yee identities for trivariate generalizations of these mock theta functions remained a mystery, as pointed out by Li and Yang in their recent work. We partially solve this problem and generalize these identities. Several new as well as well-known results are derived. For example, one of our two main theorems gives, as a corollary, a special case of Soon-Yi Kang’s three-variable reciprocity theorem. A relation between a new restricted overpartition function \(p^{*}(n)\) and a weighted partition function \(p_*(n)\) is obtained from one of the special cases of our second theorem. PubDate: 2022-06-01
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Abstract: Abstract For a semigroup S, let \({\mathcal {L}}\) and \({\mathcal {R}}\) denote the left and the right Green relation on S, respectively. Let S and T be two completely simple semigroups such that their \({\mathcal {H}}\) -classes are abelian groups. As the main result in this note, using Cameron and Ghosh’s theorem about power graphs of abelian groups, we show that S and T are isomorphic if and only if there exists a graph isomorphism \(\phi \) from the power graph of S to the power graph of T which maps an \({\mathcal {R}}\) -class and an \({\mathcal {L}}\) -class of S to an \({\mathcal {R}}\) -class and an \({\mathcal {L}}\) -class of T, respectively. PubDate: 2022-06-01
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Abstract: Abstract It is known that finite non-bipartite locally primitive arc-transitive graphs are normal covers of ‘basic objects’—vertex quasiprimitive ones. Praeger in (J London Math Soc 47(2):227–239, 1993) showed that a quasiprimitive action of a group G on a nonbipartite finite 2-arc transitive graph must be one of four of the eight O’Nan–Scott types. In this paper, we classify the basic locally primitive graphs where the action on vertices has O’Nan–Scott type \(\mathrm{HS}\) or HC, extending the well-known Praeger’s result about ‘basic’ 2-arc transitive graphs. PubDate: 2022-06-01
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Abstract: Abstract Let \(\mu _{\infty }\subseteq \mathbb {C}\) be the collection of roots of unity and \(\mathcal {C}_{n}:=\{(s_{1},\ldots ,s_{n})\in \mu _{\infty }^{n}:s_{i}\ne s_{j}\text { for any }1\le i<j\le n\}\) . Two elements \((s_{1},\ldots ,s_{n})\) and \((t_{1},\ldots ,t_{n})\) of \(\mathcal {C}_{n}\) are said to be projectively equivalent if there exists \(\gamma \in PGL (2,\mathbb {C})\) such that \(\gamma (s_{i})=t_{i}\) for any \(1\le i\le n\) . In this article, we will give a complete classification for the projectively equivalent pairs. As a consequence, we will show that the maximal length for the nontrivial projectively equivalent pairs is 14. PubDate: 2022-05-26
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Abstract: Abstract We provide an explicit construction of finite 4-regular graphs \((\Gamma _k)_{k\in {\mathbb {N}}}\) with \(\text {girth}\, \Gamma _k\rightarrow \infty \) as \(k\rightarrow \infty \) and \(\frac{\text {diam}\,\Gamma _k}{\text {girth}\,\Gamma _k}\leqslant D\) for some \(D>0\) and all \(k\in {\mathbb {N}}\) . For each fixed dimension \(n\geqslant 2,\) we find a pair of matrices in \(SL_{n}({\mathbb {Z}})\) such that (i) they generate a free subgroup, (ii) their reductions \(\bmod \, p\) generate \(SL_{n}({\mathbb {F}}_{p})\) for all sufficiently large primes p, (iii) the corresponding Cayley graphs of \(SL_{n}({\mathbb {F}}_{p})\) have girth at least \(c_n\log p\) for some \(c_n>0\) . Relying on growth results (with no use of expansion properties of the involved graphs), we observe that the diameter of those Cayley graphs is at most \(O(\log p)\) . This gives infinite sequences of finite 4-regular Cayley graphs of \(SL_n({\mathbb {F}}_p)\) as \(p\rightarrow \infty \) with large girth and bounded diameter-by-girth ratio. These are the first explicit examples in all dimensions \(n\geqslant 2\) (all prior examples were in \(n=2\) ). Moreover, they happen to be expanders. Together with Margulis’ and Lubotzky–Phillips–Sarnak’s classical constructions, these new graphs are the only known explicit logarithmic girth Cayley graph expanders. PubDate: 2022-05-22
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Abstract: Abstract Let k be a field and \(R=k[x_1,\ldots ,x_n]/I=S/I\) a graded ring. Then R has a t-linear resolution if I is generated by homogeneous elements of degree t, and all higher syzygies are linear. Thus, R has a t-linear resolution if \(\mathrm{Tor}^S_{i,j}(S/I,k)=0\) if \(j\ne i+t-1\) . For a graph G on \(\{1,\ldots ,n\}\) , the edge algebra is \(k[x_1,\ldots ,x_n]/I\) , where I is generated by those \(x_ix_j\) for which \(\{ i,j\}\) is an edge in G. We want to determine the Betti numbers of edge rings with 2-linear resolution. But we want to do that by looking at the edge ring as a Stanley–Reisner ring. For a simplicial complex \(\Delta \) on \([\mathbf{n}]=\{1,\ldots ,n\}\) and a field k, the Stanley–Reisner ring \(k[\Delta ]\) is \(k[x_1,\ldots ,x_n]/I\) , where I is generated by the squarefree monomials \(x_{i_1}\ldots x_{i_k}\) for which \(\{ i_1,\ldots ,i_k\}\) does not belong to \(\Delta \) . Which Stanley–Reisner rings that are edge rings with 2-linear resolution are known. Their associated complexes has had different names in the literature. We call them fat forests here. We determine the Betti numbers of many fat forests and compare our result with what is known. We also consider Betti numbers of Alexander duals of fat forests. PubDate: 2022-05-20
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