On Small World Semiplanes with Generalised Schubert Cells

The well known “real-life examples” of small world graphs, including the graph of binary relation: “two persons on the earth know each other” contains cliques, so they have cycles of order 3 and 4. Some problems of Computer Science require explicit construction of regular algebraic graphs with small diameter but without small cycles. The well known examples here are generalised polygons, which are small world algebraic graphs i.e. graphs with the diameter d≤clog  _k−1(v), where v is order, k is the degree and c is the independent constant, semiplanes (regular bipartite graphs without cycles of order 4); graphs that can be homomorphically mapped onto the ordinary polygons. The problem of the existence of regular graphs satisfying these conditions with the degree ≥k and the diameter ≥d for each pair k≥3 and d≥3 is addressed in the paper. This problem is positively solved via the explicit construction. Generalised Schubert cells are defined in the spirit of Gelfand-Macpherson theorem for the Grassmanian. Constructed graph, induced on the generalised largest Schubert cells, is isomorphic to the well-known Wenger’s graph. We prove that the family of edge-transitive q-regular Wenger graphs of order 2q ⁿ , where integer n≥2 and q is prime power, q≥n, q>2 is a family of small world semiplanes. We observe the applications of some classes of small world graphs without small cycles to Cryptography and Coding Theory.     

The Eulerian generating function of q-derangements

Let F_q denote the finite field with q elements. For nonnegative integers n,k, let d_q(n,k) denote the number of n×nF_q-matrices having k as the sum of the dimensions of the eigenspaces (of the eigenvalues lying in F_q). Let d_q(n)=d_q(n,0), i.e., d_q(n) denotes the number of n×nF_q-matrices having no eigenvalues in F_q. The Eulerian generating function of d_q(n) has been well studied in the last 20 years [Kung, The cycle structure of a linear transformation over a finite field, Linear Algebra Appl. 36 (1981) 141-155, Neumann and Praeger, Derangements and eigenvalue-free elements in finite classical groups, J. London Math. Soc. (2) 58 (1998) 564-586 and Stong, Some asymptotic results on finite vector spaces, Adv. Appl. Math. 9(2) (1988) 167-199]. The main tools have been the rational canonical form, nilpotent matrices, and a q-series identity of Euler. In this paper we take an elementary approach to this problem, based on Möbius inversion, and find the following bivariate generating function:

     

On tricyclic graphs of a given diameter with minimal energy

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Let G(n,d) be the class of tricyclic graphs G on n vertices with diameter d and containing no vertex disjoint odd cycles C_p,C_q of lengths p and q with p+q≡2(mod4). In this paper, we characterize the graphs with minimal energy in G(n,d).     

An explicit formula for eigenvalues of Bethe trees and upper bounds on the largest eigenvalue of any tree

A Bethe tree B_d,k is a rooted unweighted of k levels in which the root vertex has degree equal to d, the vertices at level j(2?j?k-1) have degree equal to (d+1) and the vertices at level k are the pendant vertices. In this paper, we first derive an explicit formula for the eigenvalues of the adjacency matrix of B_d,k. Moreover, we give the corresponding multiplicities. Next, we derive an explicit formula for the simple nonzero eigenvalues, among them the largest eigenvalue, of the Laplacian matrix of B_d,k. Finally, we obtain upper bounds on the largest eigenvalue of the adjacency matrix and of the Laplacian matrix of any tree T. These upper bounds are given in terms of the largest vertex degree and the radius of T, and they are attained if and only if T is a Bethe tree.     

Labeling graphs with two distance constraints

Given a graph G and integers p,q,d₁ and d₂, with p>q, d₂>d₁?1, an L(d₁,d₂;p,q)-labeling of G is a function f:V(G)→{0,1,2,…,n} such that |f(u)−f(v)|?p if d_G(u,v)?d₁ and |f(u)−f(v)|?q if d_G(u,v)?d₂. A k-L(d₁,d₂;p,q)-labeling is an L(d₁,d₂;p,q)-labeling f such that max_v∈V(G)f(v)?k. The L(d₁,d₂;p,q)-labeling number ofG, denoted by , is the smallest number k such that G has a k-L(d₁,d₂;p,q)-labeling. In this paper, we give upper bounds and lower bounds of the L(d₁,d₂;p,q)-labeling number for general graphs and some special graphs. We also discuss the L(d₁,d₂;p,q)-labeling number of G, when G is a path, a power of a path, or Cartesian product of two paths.     

The evaluation of Tornheim double sums, Part 1

We provide an explicit formula for the Tornheim double series in terms of integrals involving the Hurwitz zeta function. We also study the limit when the parameters of the Tornheim sum become natural numbers, and show that in that case it can be expressed in terms of definite integrals of triple products of Bernoulli polynomials and the Bernoulli function A_k(q)?kζ^′(1-k,q).     

On the eccentric distance sum of graphs

The eccentric distance sum is a novel topological index that offers a vast potential for structure activity/property relationships. For a graph G, it is defined as ξd(G)=_∑v∈Vε(v)D(v), where ε(v) is the eccentricity of the vertex v and D(v)=_∑u∈V(G)d(u,v) is the sum of all distances from the vertex v. Motivated by [G. Yu, L. Feng, A. Ili?, On the eccentric distance sum of trees and unicyclic graphs, J. Math. Anal. Appl. 375 (2011) 934-944], in this paper we characterize the extremal trees and graphs with maximal eccentric distance sum. Various lower and upper bounds for the eccentric distance sum in terms of other graph invariants including the Wiener index, the degree distance, eccentric connectivity index, independence number, connectivity, matching number, chromatic number and clique number are established. In addition, we present explicit formulae for the values of eccentric distance sum for the Cartesian product, applied to some graphs of chemical interest (like nanotubes and nanotori).     

Enumeration of non-crossing pairings on bit strings

A non-crossing pairing on a bit string is a matching of 1s and 0s in the string with the property that the pairing diagram has no crossings. For an arbitrary bit-string w=p₁¹q₁⁰…pr¹qr⁰, let φ(w) be the number of such pairings. This enumeration problem arises when calculating moments in the theory of random matrices and free probability, and we are interested in determining useful formulas and asymptotic estimates for φ(w). Our main results include explicit formulas in the “symmetric” case where each pi=qi, as well as upper and lower bounds for φ(w) that are uniform across all words of fixed length and fixed r. In addition, we offer more refined conjectural expressions for the upper bounds. Our proofs follow from the construction of combinatorial mappings from the set of non-crossing pairings into certain generalized “Catalan” structures that include labeled trees and lattice paths.     

Improved bounds for the Laplacian energy of Bethe trees

For k, d?2, a Bethe tree is a rooted tree with k levels which the root vertex has degree d, the vertices from level 2 to k-1 have degree d+1 and the vertices at the level k are pendent vertices. So et al, using a theorem by Ky Fan have obtained both upper and lower bounds for the Laplacian energy of bipartite graphs. We shall employ the above mentioned theorem to obtain new and improved bounds for the Laplacian energy in the case of Bethe trees.     

Nested cycles in large triangulations and crossing-critical graphs

We show that every sufficiently large plane triangulation has a large collection of nested cycles that either are pairwise disjoint, or pairwise intersect in exactly one vertex, or pairwise intersect in exactly two vertices. We apply this result to show that for each fixed positive integer k, there are only finitely many k-crossing-critical simple graphs of average degree at least six. Combined with the recent constructions of crossing-critical graphs given by Bokal, this settles the question of for which numbers q>0 there is an infinite family of k-crossing-critical simple graphs of average degree q.     

Bounds on some van der Waerden numbers

For positive integers s and k₁,k₂,…,ks, the van der Waerden number w(k₁,k₂,…,ks;s) is the minimum integer n such that for every s-coloring of set {1,2,…,n}, with colors 1,2,…,s, there is a ki-term arithmetic progression of color i for some i. We give an asymptotic lower bound for w(k,m;2) for fixed m. We include a table of values of w(k,3;2) that are very close to this lower bound for m=3. We also give a lower bound for w(k,k,…,k;s) that slightly improves previously-known bounds. Upper bounds for w(k,4;2) and w(4,4,…,4;s) are also provided.     

Finding small regular graphs of girths 6, 8 and 12 as subgraphs of cages

Small k-regular graphs of girth g where g=6,8,12 are obtained as subgraphs of minimal cages. More precisely, we obtain (k,6)-graphs on 2(kq−1) vertices, (k,8)-graphs on 2k(q²−1) vertices and (k,12)-graphs on 2kq²(q²−1), where q is a prime power and k is a positive integer such that q≥k≥3. Some of these graphs have the smallest number of vertices known so far among the regular graphs with girth g=6,8,12.     

Graphs of order two less than the Moore bound

The problem of determining the largest order n_d,k of a graph of maximum degree at most d and diameter at most k is well known as the degree/diameter problem. It is known that n_d,k?M_d,k where M_d,k is the Moore bound. For d=4, the current best upper bound for n_4,k is M_4,k-1. In this paper we study properties of graphs of order M_d,k-2 and we give a new upper bound for n_4,k for k?3.     

On anti-magic labeling for graph products

An anti-magic labeling of a finite simple undirected graph with p vertices and q edges is a bijection from the set of edges to the set of integers {1,2,…,q} such that the vertex sums are pairwise distinct, where the vertex sum at one vertex is the sum of labels of all edges incident to such vertex. A graph is called anti-magic if it admits an anti-magic labeling. Hartsfield and Ringel conjectured in 1990 that all connected graphs except K₂ are anti-magic. Recently, Alon et al. showed that this conjecture is true for dense graphs, i.e. it is true for p-vertex graphs with minimum degree Ω(logp). In this article, new classes of sparse anti-magic graphs are constructed through Cartesian products and lexicographic products.     

The d-precoloring problem for k-degenerate graphs

In this paper we deal with the d-PRECOLORING EXTENSION (d-PREXT) problem in various classes of graphs. The d-PREXT problem is the special case of PRECOLORING EXTENSION problem where, for a fixed constant d, input instances are restricted to contain at most d precolored vertices for every available color. The goal is to decide if there exists an extension of given precoloring using only available colors or to find it.We present a linear time algorithm for both, the decision and the search version of d-PREXT, in the following cases: (i) restricted to the class of k-degenerate graphs (hence also planar graphs) and with sufficiently large set S of available colors, and (ii) restricted to the class of partial k-trees (without any size restriction on S). We also study the following problem related to d-PREXT: given an instance of the d-PREXT problem which is extendable by colors of S, what is the minimum number of colors of S sufficient to use for precolorless vertices over all such extensions? We establish lower and upper bounds on this value for k-degenerate graphs and its various subclasses (e.g., planar graphs, outerplanar graphs) and prove tight results for the class of trees.     

New linear codes over GF(8)

Let [n, k, d; q]-codes be linear codes of length n, dimension k and minimum Hamming distance d over GF(q). Let d₈(n, k) be the maximum possible minimum Hamming distance of a linear [n, k, d; 8]-code for given values of n and k. In this paper, eighteen new linear codes over GF(8) are constructed which improve the table of d₈(n, k) by Brouwer.     

Breaking the rhythm on graphs

We study graph colorings avoiding periodic sequences with large number of blocks on paths. The main problem is to decide, for a given class of graphs F, if there are absolute constants t,k such that any graph from the class has a t-coloring with no k identical blocks in a row appearing on a path. The minimum t for which there is some k with this property is called the rhythm threshold of F, denoted by t(F). For instance, we show that the rhythm threshold of graphs of maximum degree at most d is between (d+1)/2 and d+1. We give several general conditions for finiteness of t(F), as well as some connections to existing chromatic parameters. The question whether the rhythm threshold is finite for planar graphs remains open.     

Trees of extremal connectivity index

The connectivity index w_α(G) of a graph G is the sum of the weights (d(u)d(v))^α of all edges uv of G, where α is a real number (α≠0), and d(u) denotes the degree of the vertex u. Let T be a tree with n vertices and k pendant vertices. In this paper, we give sharp lower and upper bounds for w₁(T). Also, for -1?α<0, we give a sharp lower bound and a upper bound for w_α(T).     

On the Extendability of Linear Codes

R. Hill and P. Lizak (1995, in “Proc. IEEE Int. Symposium on Inform. Theory, Whistler, Canada,” pp. 345) proved that every [n, k, d]_q code with gcd(d, q)=1 and with all weights congruent to 0 or d (modulo q) is extendable to an [n+1, k, d+1]_q code with all weights congruent to 0 or d+1 (modulo q). We give another elementary geometrical proof of this theorem, which also yields the uniqueness of the extension.     

Product decompositions of the symmetric group induced by separable permutations

In this paper we consider the rank generating function of a separable permutation π in the weak Bruhat order on the two intervals [id,π] and [π,w₀], where w₀=n,n−1,…,1. We show a surprising result that the product of these two generating functions is the generating function for the symmetric group with the weak order. We then obtain explicit formulas for the rank generating functions on [id,π] and [π,w₀], leading to the rank-symmetry and unimodality of the two graded posets.