On matching cover of graphs

A k-matching cover of a graph \(G\) is a union of \(k\) matchings of \(G\) which covers \(V(G)\) . The matching cover number of \(G\) , denoted by \(mc(G)\) , is the minimum number \(k\) such that \(G\) has a \(k\) -matching cover. A matching cover of \(G\) is optimal if it consists of \(mc(G)\) matchings of \(G\) . In this paper, we present an algorithm for finding an optimal matching cover of a graph on \(n\) vertices in \(O(n^3)\) time (if use a faster maximum matching algorithm, the time complexity can be reduced to \(O(nm)\) , where \(m=|E(G)|\) ), and give an upper bound on matching cover number of graphs. In particular, for trees, a linear-time algorithm is given, and as a by-product, the matching cover number of trees is determined.     

On Sub-determinants and the Diameter of Polyhedra

We derive a new upper bound on the diameter of a polyhedron \(P = \{x {\in } {\mathbb {R}}^n :Ax\le b\}\) , where \(A \in {\mathbb {Z}}^{m\times n}\) . The bound is polynomial in \(n\) and the largest absolute value of a sub-determinant of \(A\) , denoted by \(\Delta \) . More precisely, we show that the diameter of \(P\) is bounded by \(O(\Delta ^2 n^4\log n\Delta )\) . If \(P\) is bounded, then we show that the diameter of \(P\) is at most \(O(\Delta ^2 n^{3.5}\log n\Delta )\) . For the special case in which \(A\) is a totally unimodular matrix, the bounds are \(O(n^4\log n)\) and \(O(n^{3.5}\log n)\) respectively. This improves over the previous best bound of \(O(m^{16}n^3(\log mn)^3)\) due to Dyer and Frieze (Math Program 64:1–16, 1994).     

Scattered linear sets generated by collineations between pencils of lines

Let \(A\) and \(B\) be two points of \(\mathrm{{PG}}(2,q^n)\) , and let \(\Phi \) be a collineation between the pencils of lines with vertices \(A\) and \(B\) . In this paper, we prove that the set of points of intersection of corresponding lines under \(\Phi \) is either the union of a scattered \(\mathrm{{GF}}(q)\) -linear set of rank \(n+1\) with the line \(AB\) or the union of \(q-1\) scattered \(\mathrm{{GF}}(q)\) -linear sets of rank \(n\) with \(A\) and \(B\) . We also determine the intersection configurations of two scattered \(\mathrm{{GF}}(q)\) -linear sets of rank \(n+1\) of \(\mathrm{{PG}}(2,q^n)\) both meeting the line \(AB\) in a \(\mathrm{{GF}}(q)\) -linear set of pseudoregulus type with transversal points \(A\) and \(B\) .     

Additive mappings satisfying algebraic conditions in rings

Let \(R\) be any \((n+1)!\) -torsion free ring and \(F,D: R\rightarrow R\) be additive mappings satisfying \(F(x^{n+1})=(\alpha (x))^nF(x)+\sum \nolimits _{i=1}^n (\alpha (x))^{n-i}(\beta (x))^iD(x)\) for all \(x\in R\) , where \(n\) is a fixed integer and \(\alpha \) , \(\beta \) are automorphisms of \(R\) . Then, \(D\) is Jordan left \((\alpha , \beta )\) -derivation and \(F\) is generalized Jordan left \((\alpha , \beta )\) -derivation on \(R\) and if additive mappings \(F\) and \(D\) satisfying \(F(x^{n+1})=F(x)(\alpha (x))^n+\sum \nolimits _{i=1}^n (\beta (x))^iD(x)(\alpha (x))^{n-i}\) for all \(x\in R\) . Then, \(D\) is Jordan \((\alpha , \beta )\) -derivation and \(F\) is generalized Jordan \((\alpha , \beta )\) -derivation on \(R\) . At last some immediate consequences of the above theorems have been given.     

SFT-stability and Krull dimension in power series rings over an almost pseudo-valuation domain

Let \(R\) be an APVD with maximal ideal \(M\) . We show that the power series ring \(R[[x_1,\ldots ,x_n]]\) is an SFT-ring if and only if the integral closure of \(R\) is an SFT-ring if and only if ( \(R\) is an SFT-ring and \(M\) is a Noether strongly primary ideal of \((M:M)\) ). We deduce that if \(R\) is an \(m\) -dimensional APVD that is a residually *-domain, then dim \(R[[x_1,\ldots ,x_n]]\,=\,nm+1\) or \(nm+n\) .     

On the rank of incidence matrices in projective Hjelmslev spaces

Let \(R\) be a finite chain ring with \(|R|=q^m\) , \(R/{{\mathrm{Rad}}}R\cong \mathbb {F}_q\) , and let \(\Omega ={{\mathrm{PHG}}}({}_RR^n)\) . Let \(\tau =(\tau _1,\ldots ,\tau _n)\) be an integer sequence satisfying \(m=\tau _1\ge \tau _2\ge \cdots \ge \tau _n\ge 0\) . We consider the incidence matrix of all shape \(\varvec{m}^s=(\underbrace{m,\ldots ,m}_s)\) versus all shape \(\tau \) subspaces of \(\Omega \) with \(\varvec{m}^s\preceq \tau \preceq \varvec{m}^{n-s}\) . We prove that the rank of \(M_{\varvec{m}^s,\tau }(\Omega )\) over \(\mathbb {Q}\) is equal to the number of shape \(\varvec{m}^s\) subspaces. This is a partial analog of Kantor’s result about the rank of the incidence matrix of all \(s\) dimensional versus all \(t\) dimensional subspaces of \({{\mathrm{PG}}}(n,q)\) . We construct an example for shapes \(\sigma \) and \(\tau \) for which the rank of \(M_{\sigma ,\tau }(\Omega )\) is not maximal.     

L^p-Integrability,Dimensions of Supports of Fourier Transforms and Applications

It is proved that there does not exist any non zero function in \(L^p({\mathbb R}^n)\) with \(1\le p\le 2n/\alpha \) if its Fourier transform is supported by a set of finite packing \(\alpha \) -measure where \(0<\alpha <n\) . It is shown that the assertion fails for \(p>2n/\alpha \) . The result is applied to prove \(L^p\) Wiener Tauberian theorems for \({\mathbb R}^n\) and \(M(2)\) .     

A groupoid generalisation of Leavitt path algebras

Let \(G\) be a locally compact, Hausdorff, étale groupoid whose unit space is totally disconnected. We show that the collection \(A(G)\) of locally-constant, compactly supported complex-valued functions on \(G\) is a dense \(*\) -subalgebra of \(C_c(G)\) and that it is universal for algebraic representations of the collection of compact open bisections of \(G\) . We also show that if \(G\) is the groupoid associated to a row-finite graph or \(k\) -graph with no sources, then \(A(G)\) is isomorphic to the associated Leavitt path algebra or Kumjian–Pask algebra. We prove versions of the Cuntz–Krieger and graded uniqueness theorems for \(A(G)\) .     

Counting and Sampling Minimum Cuts in Genus g Graphs

Let \(G\) be a directed graph with \(n\) vertices embedded on an orientable surface of genus \(g\) with two designated vertices \(s\) and \(t\) . We show that computing the number of minimum \((s,t)\) -cuts in \(G\) is fixed-parameter tractable in \(g\) . Specifically, we give a \(2^{O(g)} n^2\) time algorithm for this problem. Our algorithm requires counting sets of cycles in a particular integer homology class. That we can count these cycles is an interesting result in itself as there are no prior results that are fixed-parameter tractable and deal directly with integer homology. We also describe an algorithm which, after running our algorithm to count minimum cuts once, can sample an \((s,t)\) -minimum cut uniformly at random in \(O(n \log n)\) time per sample.     

Pointwise convergence in Pringsheim’s sense of the summability of Fourier transforms on Wiener amalgam spaces

New multi-dimensional Wiener amalgam spaces \(W_c(L_p,\ell _\infty )(\mathbb{R }^d)\) are introduced by taking the usual one-dimensional spaces coordinatewise in each dimension. The strong Hardy-Littlewood maximal function is investigated on these spaces. The pointwise convergence in Pringsheim’s sense of the \(\theta \) -summability of multi-dimensional Fourier transforms is studied. It is proved that if the Fourier transform of \(\theta \) is in a suitable Herz space, then the \(\theta \) -means \(\sigma _T^\theta f\) converge to \(f\) a.e. for all \(f\in W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d)\) . Note that \(W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d) \supset W_c(L_r,\ell _\infty )(\mathbb{R }^d) \supset L_r(\mathbb{R }^d)\) and \(W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d) \supset L_1(\log L)^{d-1}(\mathbb{R }^d)\) , where \(1 . Moreover, \(\sigma _T^\theta f(x)\) converges to \(f(x)\) at each Lebesgue point of \(f\in W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d)\) .     

Proper circulant weighing matrices of weight p^2

A circulant weighing matrix \(CW(v,n)\) is a circulant matrix \(M\) of order \(v\) with \(0,\pm 1\) entries such that \(MM^T=nI_v\) . In this paper, we study proper circulant matrices with \(n=p^2\) where \(p\) is an odd prime divisor of \(v\) . For \(p\ge 5\) , it turns out that to search for such circulant matrices leads us to two group ring equations and by studying these two equations, we manage to prove that no proper \(CW(pw,p^2)\) exists when \(p\equiv 3\pmod {4}\) or \(p=5\) .     

Vertex transitive graphs from injective linear mappings

Based on a motivation coming from the study of the metric structure of the category of finite dimensional vector spaces over a finite field \(\mathbb {F}\) , we examine a family of graphs, defined for each pair of integers \(1 \le k \le n\) , with vertex set formed by all injective linear transformations \(\mathbb {F}^k \rightarrow \mathbb {F}^n\) and edges corresponding to pairs of mappings, \(f\) and \(g\) , with \(\lambda (f,g)= \dim \mathrm{Im }(f-g)=1 \) . For \(\mathbb {F}\cong \mathrm{GF }(q)\) , this graph will be denoted by \(\mathrm{INJ }_q(k,n)\) . We show that all such graphs are vertex transitive and Hamiltonian and describe the full automorphism group of each \(\mathrm{INJ }_q (k,n)\) for \(k . Using the properties of line-transitive groups, we completely determine which of the graphs \(\mathrm{INJ }_q (k,n)\) are Cayley and which are not. The Cayley ones consist of three infinite families, corresponding to pairs \((1,n),\,(n-1,n)\) , and \((n,n)\) , with \(n\) and \(q\) arbitrary, and of two sporadic examples \(\mathrm{INJ }_{2} (2,5)\) and \(\mathrm{INJ }_{2}(3,5)\) . Hence, the overwhelming majority of our graphs is not Cayley.     

Endomorphism rings of bimodules

Let \(M\) be an \(R\) - \(R\) -bimodule over a semi-prime right and left Goldie ring \(R\) . We investigate how non-singularity conditions on \(M_R\) are related to such conditions on \(_RM\) . In particular, we say an \(R\) - \(R\) -bimodule \(M\) such that \(_RM\) and \(M_R\) are non-singular has the right essentiality property if \(IM_R\) is essential in \(M_R\) for all essential right ideals \(I\) of \(R\) , and investigate several questions related to this property.     

t-Structures and cotilting modules over commutative noetherian rings

For a commutative noetherian ring \(R\) , we establish a bijection between the resolving subcategories consisting of finitely generated \(R\) -modules of finite projective dimension and the compactly generated t-structures in the unbounded derived category \(\mathcal {D}(R)\) that contain \(R[1]\) in their heart. Under this bijection, the t-structures \((\mathcal U,\mathcal V)\) such that the aisle \(\mathcal U\) consists of objects with homology concentrated in degrees \(<n\) correspond to the \(n\) -cotilting classes in \({{\mathrm{Mod}\text {-}R}}\) . As a consequence of these results, we prove that the little finitistic dimension findim \(R\) of \(R\) equals an integer \(n\) if and only if the direct sum \(\bigoplus _{k=0}^n E_k(R)\) of the first \(n+1\) terms in a minimal injective coresolution \(0\rightarrow R\rightarrow E_0(R)\rightarrow E_1(R)\rightarrow \cdots \) of \(R\) is an injective cogenerator of \({{\mathrm{Mod}\text {-}R}}\) .     

Isomorphisms between Jacobson graphs

Let \(R\) be a commutative ring with a non-zero identity and \(\mathfrak {J}_R\) be its Jacobson graph. We show that if \(R\) and \(R'\) are finite commutative rings, then \(\mathfrak {J}_R\cong \mathfrak {J}_{R'}\) if and only if \(|J(R)|=|J(R')|\) and \(R/J(R)\cong R'/J(R')\) . Also, for a Jacobson graph \(\mathfrak {J}_R\) , we obtain the structure of group \(\mathrm {Aut}(\mathfrak {J}_R)\) of all automorphisms of \(\mathfrak {J}_R\) and prove that under some conditions two semi-simple rings \(R\) and \(R'\) are isomorphic if and only if \(\mathrm {Aut}(\mathfrak {J}_R)\cong \mathrm {Aut}(\mathfrak {J}_{R'})\) .     

Neighbour-transitive codes in Johnson graphs

The Johnson graph \(J(v,k)\) has, as vertices, the \(k\) -subsets of a \(v\) -set \(\mathcal {V}\) and as edges the pairs of \(k\) -subsets with intersection of size \(k-1\) . We introduce the notion of a neighbour-transitive code in \(J(v,k)\) . This is a proper vertex subset \(\Gamma \) such that the subgroup \(G\) of graph automorphisms leaving \(\Gamma \) invariant is transitive on both the set \(\Gamma \) of ‘codewords’ and also the set of ‘neighbours’ of \(\Gamma \) , which are the non-codewords joined by an edge to some codeword. We classify all examples where the group \(G\) is a subgroup of the symmetric group \(\mathrm{Sym}\,(\mathcal {V})\) and is intransitive or imprimitive on the underlying \(v\) -set \(\mathcal {V}\) . In the remaining case where \(G\le \mathrm{Sym}\,(\mathcal {V})\) and \(G\) is primitive on \(\mathcal {V}\) , we prove that, provided distinct codewords are at distance at least \(3\) , then \(G\) is \(2\) -transitive on \(\mathcal {V}\) . We examine many of the infinite families of finite \(2\) -transitive permutation groups and construct surprisingly rich families of examples of neighbour-transitive codes. A major unresolved case remains.     

Low Complexity Methods For Discretizing Manifolds Via Riesz Energy Minimization

Let \(A\) be a compact \(d\) -rectifiable set embedded in Euclidean space \({\mathbb R}^p, d\le p\) . For a given continuous distribution \(\sigma (x)\) with respect to a \(d\) -dimensional Hausdorff measure on \(A\) , our earlier results provided a method for generating \(N\) -point configurations on \(A\) that have an asymptotic distribution \(\sigma (x)\) as \(N\rightarrow \infty \) ; moreover, such configurations are “quasi-uniform” in the sense that the ratio of the covering radius to the separation distance is bounded independently of \(N\) . The method is based upon minimizing the energy of \(N\) particles constrained to \(A\) interacting via a weighted power-law potential \(w(x,y)|x-y|^{-s}\) , where \(s>d\) is a fixed parameter and \(w(x,y)=\left( \sigma (x)\sigma (y)\right) ^{-({s}/{2d})}\) . Here we show that one can generate points on \(A\) with the aforementioned properties keeping in the energy sums only those pairs of points that are located at a distance of at most \(r_N=C_N N^{-1/d}\) from each other, with \(C_N\) being a positive sequence tending to infinity arbitrarily slowly. To do this, we minimize the energy with respect to a varying truncated weight \(v_N(x,y)=\Phi (|x-y|/r_N)\cdot w(x,y)\) , where \(\Phi :(0,\infty )\rightarrow [0,\infty )\) is a bounded function with \(\Phi (t)=0, t\ge 1\) , and \(\lim _{t\rightarrow 0^+}\Phi (t)=1\) . Under appropriate assumptions, this reduces the complexity of generating \(N\) -point “low energy” discretizations to order \(N C_N^d\) computations.     

On the Chromatic Number of \mathbb {R}^4

The lower bound for the chromatic number of \(\mathbb {R}^4\) is improved from \(7\) to \(9\) . Three graphs with unit distance embeddings in \(\mathbb {R}^4\) are described. The first is a \(7\) -chromatic graph of order \(14\) whose chromatic number can be verified by inspection. The second is an \(8\) -chromatic graph of order \(26\) . In this case the chromatic number can be verified quickly by a simple computer program. The third graph is a \(9\) -chromatic graph of order \(65\) for which computer verification takes about one minute.