The number of reachable pairs in a digraph

For a digraph G, let R(G) (respectively, R^(k)(G)) be the number of ordered pairs (u,v) of vertices of G such that u≠v and v is reachable from u (respectively, reachable from u by a path of length ?k). In this paper, we study the range S_n of R(G) and the range of R^(k)(G) as G varies over all possible digraphs on n vertices. We give a sufficient condition and a necessary condition for an integer to belong to S_n. These determine the set S_n for all n?208. We also determine for k?4 and show that whenever n?k+(k+1)^0.57+2, for arbitrary k.     

Kernels of Gramian operators for frames in shift-invariant subspaces

For an invertible n×n matrix B and Φ a finite or countable subset of L²(Rⁿ), consider the collection X={?(·-Bk):?∈Φ,k∈Zⁿ} generating the closed subspace M of L²(Rⁿ). Our main objects of interest in this paper are the kernel of the associated Gramian G(.) and dual Gramian operator-valued functions. We show in particular that the orthogonal complement of M in L²(Rⁿ) can be generated by a Parseval frame obtained from a shift-invariant system having m generators where . Furthermore, this Parseval frame can be taken to be an orthonormal basis exactly when almost everywhere. Analogous results in terms of dim(Ker(G(.))) are also obtained concerning the existence of a collection of m sequences in the orthogonal complement of the range of analysis operator associated with the frame X whose shifts either form a Parseval frame or an orthonormal basis for that orthogonal complement.     

Eigenvalues and degree deviation in graphs

Let G be a graph with n vertices and m edges and let μ(G) = μ₁(G) ? ? ? μ_n(G) be the eigenvalues of its adjacency matrix. Set s(G)=∑_u∈V(G)∣d(u)-2m/n∣. We prove that

     

Divisors of the number of Latin rectangles

A k×n Latin rectangle on the symbols {1,2,…,n} is called reduced if the first row is (1,2,…,n) and the first column is T(1,2,…,k). Let Rk,n be the number of reduced k×n Latin rectangles and m=⌊n/2⌋. We prove several results giving divisors of Rk,n. For example, (k−1)! divides Rk,n when k?m and m! divides Rk,n when m<k?n. We establish a recurrence which determines the congruence class of for a range of different t. We use this to show that Rk,n≡((−1)^k−1(k−1)!)ⁿ⁻¹. In particular, this means that if n is prime, then Rk,n≡1 for 1?k?n and if n is composite then if and only if k is larger than the greatest prime divisor of n.     

Moment conditions and support theorems for Radon transforms on affine Grassmann manifolds

Let G(p,n) and G(q,n) be the affine Grassmann manifolds of p- and q-planes in Rⁿ, respectively, and let be the Radon transform from smooth functions on G(p,n) to smooth functions on G(q,n) arising from the inclusion incidence relation. When p<q and dimG(p,n)=dimG(p,n), we present a range characterization theorem for via moment conditions. We then use this range result to prove a support theorem for . This complements a previous range characterization theorem for via differential equations when dimG(p,n)<dimG(p,n). We also present a support theorem in this latter case.     

Hankel planes

Starting from an (m+1)×(n+1) matrix A one can construct (m+p+1)×(n+1)(p+1) block Toeplitz matrices , p≥0, based on the rows of A. The connections between the ranks of the two matrices is studied by comparing the corresponding vector spaces of row relations R and R(p). A main tool are the Hankel matrices with rows in R. The dimension of R(p) is determined in terms of geometric invariants attached to the Hankel matrices with rows in R. The study of Hankel r-planes of P^m, for r≥1, turns out to be very useful and interesting in itself since they constitute a subvariety of the Grassmannian G(r,m).     

Representations of matroids and free resolutions for multigraded modules

Let k be a field, let R=k[x₁,…,xm] be a polynomial ring with the standard Zm-grading (multigrading), let L be a Noetherian multigraded R-module, and let be a finite free multigraded presentation of L over R. Given a choice S of a multihomogeneous basis of E, we construct an explicit canonical finite free multigraded resolution T_•(Φ,S) of the R-module L. In the case of monomial ideals our construction recovers the Taylor resolution. A main ingredient of our work is a new linear algebra construction of independent interest, which produces from a representation ? over k of a matroid M a canonical finite complex of finite dimensional k-vector spaces T_•(?) that is a resolution of Ker?. We also show that the length of T_•(?) and the dimensions of its components are combinatorial invariants of the matroid M, and are independent of the representation map ?.     

Note on the energy of regular graphs

For a simple graph G, the energy E(G) is defined as the sum of the absolute values of all the eigenvalues of its adjacency matrix A(G). Let n,m, respectively, be the number of vertices and edges of G. One well-known inequality is that , where λ₁ is the spectral radius. If G is k-regular, we have . Denote . Balakrishnan [R. Balakrishnan, The energy of a graph, Linear Algebra Appl. 387 (2004) 287-295] proved that for each ?>0, there exist infinitely many n for each of which there exists a k-regular graph G of order n with k<n-1 and , and proposed an open problem that, given a positive integer n?3, and ?>0, does there exist a k-regular graph G of order n such that . In this paper, we show that for each ?>0, there exist infinitely many such n that . Moreover, we construct another class of simpler graphs which also supports the first assertion that .     

The Ramsey numbers for disjoint unions of trees

For given graphs G and H, the Ramsey numberR(G,H) is the smallest natural number n such that for every graph F of order n: either F contains G or the complement of F contains H. In this paper, we investigate the Ramsey number R(∪G,H), where G is a tree and H is a wheel W_m or a complete graph K_m. We show that if n?3, then R(kS_n,W₄)=(k+1)n for k?2, even n and R(kS_n,W₄)=(k+1)n-1 for k?1 and odd n. We also show that .     

The Ramsey numbers for disjoint unions of graphs

For given graphs G and H, the Ramsey number R(G,H) is the smallest natural number n such that for every graph F of order n: either F contains G or the complement of F contains H. In this paper we investigate the Ramsey number of a disjoint union of graphs . For any natural integer k, we contain a general upper bound, R(kG,H)?R(G,H)+(k-1)|V(G)|. We also show that if m=2n-4, 2n-8 or 2n-6, then R(kS_n,W_m)=R(S_n,W_m)+(k-1)n. Furthermore, if |G_i|>(|G_i|-|G_i+1|)(χ(H)-1) and R(G_i,H)=(χ(H)-1)(|G_i|-1)+1, for each i, then .     

Stable extendibility of normal bundles over lens spaces

We study the stable extendibility of R-vector bundles over the (2n+1)-dimensional standard lens space Ln(p) with odd prime p, focusing on the normal bundle to an immersion of Ln(p) in the Euclidean space R2n+1+t. We show several concrete cases in which is stably extendible to Lk(p) for any k with k?n, and in several cases we determine the exact value m for which is stably extendible to Lm(p) but not stably extendible to Lm+1(p).     

Some relations between rank of a graph and its complement

Let G be a graph of order n and rank(G) denotes the rank of its adjacency matrix. Clearly, . In this paper we characterize all graphs G such that or n + 2. Also for every integer n ? 5 and any k, 0 ? k ? n, we construct a graph G of order n, such that .     

Roman domination in regular graphs

A Roman domination function on a graph G=(V(G),E(G)) is a function f:V(G)→{0,1,2} satisfying the condition that every vertex u for which f(u)=0 is adjacent to at least one vertex v for which f(v)=2. The weight of a Roman dominating function is the value f(V(G))=∑_u∈V(G)f(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. Cockayne et al. [E. J. Cockayne et al. Roman domination in graphs, Discrete Mathematics 278 (2004) 11-22] showed that γ(G)≤γ_R(G)≤2γ(G) and defined a graph G to be Roman if γ_R(G)=2γ(G). In this article, the authors gave several classes of Roman graphs: P_3k,P_3k+2,C_3k,C_3k+2 for k≥1, K_m,n for min{m,n}≠2, and any graph G with γ(G)=1; In this paper, we research on regular Roman graphs and prove that: (1) the circulant graphs and , n⁄≡1 (mod (2k+1)), (n≠2k) are Roman graphs, (2) the generalized Petersen graphs P(n,2k+1)( (mod 4) and ), P(n,1) (n⁄≡2 (mod 4)), P(n,3) ( (mod 4)) and P(11,3) are Roman graphs, and (3) the Cartesian product graphs are Roman graphs.     

Complexity results in graph reconstruction

We investigate the relative complexity of the graph isomorphism problem (GI) and problems related to the reconstruction of a graph from its vertex-deleted or edge-deleted subgraphs (in particular, deck checking (DC) and legitimate deck (LD) problems). We show that these problems are closely related for all amounts c?1 of deletion:

(1)

, , , and .

(2)

For all k?2, and .

(3)

For all k?2, .

(4)

.

(5)

For all k?2, .

For many of these results, even the c=1 case was not previously known.Similar to the definition of reconstruction numbers vrn_∃(G) [F. Harary, M. Plantholt, The graph reconstruction number, J. Graph Theory 9 (1985) 451-454] and ern_∃(G) (see [J. Lauri, R. Scapellato Topics in Graph Automorphism and Reconstruction, London Mathematical Society, Cambridge University Press, Cambridge, 2003, p. 120]), we introduce two new graph parameters, vrn_∀(G) and ern_∀(G), and give an example of a family {G_n}_n?4 of graphs on n vertices for which vrn_∃(G_n)<vrn_∀(G_n). For every k?2 and n?1, we show that there exists a collection of k graphs on (2^k-1+1)n+k vertices with 2ⁿ 1-vertex-preimages, i.e., one has families of graph collections whose number of 1-vertex-preimages is huge relative to the size of the graphs involved.     

Graphs with maximal signless Laplacian spectral radius

By the signless Laplacian of a (simple) graph G we mean the matrix Q(G)=D(G)+A(G), where A(G),D(G) denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of G. It is known that connected graphs G that maximize the signless Laplacian spectral radius ρ(Q(G)) over all connected graphs with given numbers of vertices and edges are (degree) maximal. For a maximal graph G with n vertices and r distinct vertex degrees δ_r>δ_r-1>?>δ₁, it is proved that ρ(Q(G))<ρ(Q(H)) for some maximal graph H with n+1 (respectively, n) vertices and the same number of edges as G if either G has precisely two dominating vertices or there exists an integer such that δ_i+δ_r+1-i?n+1 (respectively, δ_i+δ_r+1-i?δ_l+δ_r-l+1). Graphs that maximize ρ(Q(G)) over the class of graphs with m edges and m-k vertices, for k=0,1,2,3, are completely determined.     

Cubicity, boxicity, and vertex cover

A k-dimensional box is the cartesian product R₁×R₂×?×R_k where each R_i is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer k such that G is the intersection graph of a collection of k-dimensional boxes. A unit cube in k-dimensional space or a k-cube is defined as the cartesian product R₁×R₂×?×R_k where each R_i is a closed interval on the real line of the form [a_i,a_i+1]. The cubicity of G, denoted as cub(G), is the minimum k such that G is the intersection graph of a collection of k-cubes. In this paper we show that cub(G)≤t+⌈log(n−t)⌉−1 and , where t is the cardinality of a minimum vertex cover of G and n is the number of vertices of G. We also show the tightness of these upper bounds.F.S. Roberts in his pioneering paper on boxicity and cubicity had shown that for a graph G, and , where n is the number of vertices of G, and these bounds are tight. We show that if G is a bipartite graph then and this bound is tight. We also show that if G is a bipartite graph then . We point out that there exist graphs of very high boxicity but with very low chromatic number. For example there exist bipartite (i.e., 2 colorable) graphs with boxicity equal to . Interestingly, if boxicity is very close to , then chromatic number also has to be very high. In particular, we show that if , s≥0, then , where χ(G) is the chromatic number of G.     

Homogeneous polynomials as Lyapunov functions in the stability research of solutions of difference equations

The linear autonomous system of difference equations x(n+1)=Ax(n) is considered, where is a real nonsingular k×k matrix. In this paper it has been proved that if W(x) is any homogeneous polynomial of m-th degree in x, then there exists a unique homogeneous polynomial V(x) of m-th degree such that ΔV=V(Ax)-V(x)=W(x) if and only if where are the eigenvalues of the matrix A. The theorem on the instability has also been proved.     

On some three color Ramsey numbers for paths and cycles

For given graphs G₁,G₂,…,G_k, k≥2, the multicolor Ramsey number, denoted by R(G₁,G₂,…,G_k), is the smallest integer n such that if we arbitrarily color the edges of a complete graph on n vertices with k colors, there is always a monochromatic copy of G_i colored with i, for some 1≤i≤k. Let P_k (resp. C_k) be the path (resp. cycle) on k vertices. In the paper we consider the value for numbers of type R(P_i,P_k,C_m) for odd m, k≥m≥3 and when i is odd, and when i is even. In addition, we provide the exact values for Ramsey numbers R(P₃,P_k,C₄) for all integers k≥3.     

The hardness of approximating the boxicity, cubicity and threshold dimension of a graph

A k-dimensional box is the Cartesian product R₁×R₂×?×R_k where each R_i is a closed interval on the real line. The boxicity of a graph G, denoted as , is the minimum integer k such that G can be represented as the intersection graph of a collection of k-dimensional boxes. A unit cube in k-dimensional space or a k-cube is defined as the Cartesian product R₁×R₂×?×R_k where each R_i is a closed interval on the real line of the form [a_i,a_i+1]. The cubicity of G, denoted as , is the minimum integer k such that G can be represented as the intersection graph of a collection of k-cubes. The threshold dimension of a graph G(V,E) is the smallest integer k such that E can be covered by k threshold spanning subgraphs of G. In this paper we will show that there exists no polynomial-time algorithm for approximating the threshold dimension of a graph on n vertices with a factor of O(n^0.5−?) for any ?>0 unless NP=ZPP. From this result we will show that there exists no polynomial-time algorithm for approximating the boxicity and the cubicity of a graph on n vertices with factor O(n^0.5−?) for any ?>0 unless NP=ZPP. In fact all these hardness results hold even for a highly structured class of graphs, namely the split graphs. We will also show that it is NP-complete to determine whether a given split graph has boxicity at most 3.