Edge-recognizable domination numbers

For any undirected graph G, let be the collection of edge-deleted subgraphs. It is always possible to construct a graph H from so that . The general edge-reconstruction conjecture states that G and H must be isomorphic if they have at least four edges. A graphical invariant that must be identical for all graphs that can be constructed from the given collection is said to be edge-recognizable. Here we show that the domination number and many of its common variations are edge-recognizable.     

Lower bounds on several versions of signed domination number

Three numerical invariants of graphs concerning domination, which are named the signed domination number γ_s, the k-subdomination number γ_ks and the signed total domination number γ_st, are studied in this paper. For any graph, some lower bounds on γ_s, γ_ks and γ_st are presented, some of which generalize several known lower bounds on γ_s, γ_ks and γ_st, while others are considered as new. It is also shown that these bounds are sharp.     

Total minus domination in k-partite graphs

A function f defined on the vertices of a graph G=(V,E),f:V→{-1,0,1} is a total minus dominating function (TMDF) if the sum of its values over any open neighborhood is at least one. The weight of a TMDF is the sum of its function values over all vertices. The total minus domination number, denoted by , of G is the minimum weight of a TMDF on G. In this paper, a sharp lower bound on of k-partite graphs is given.     

On low degree k-ordered graphs

A simple graph G is k-ordered (respectively, k-ordered hamiltonian) if, for any sequence of k distinct vertices v₁,…,v_k of G, there exists a cycle (respectively, a hamiltonian cycle) in G containing these k vertices in the specified order. In 1997 Ng and Schultz introduced these concepts of cycle orderability, and motivated by the fact that k-orderedness of a graph implies (k-1)-connectivity, they posed the question of the existence of low degree k-ordered hamiltonian graphs. We construct an infinite family of graphs, which we call bracelet graphs, that are (k-1)-regular and are k-ordered hamiltonian for odd k. This result provides the best possible answer to the question of the existence of low degree k-ordered hamiltonian graphs for odd k. We further show that for even k, there exist no k-ordered bracelet graphs with minimum degree k-1 and maximum degree less than k+2, and we exhibit an infinite family of bracelet graphs with minimum degree k-1 and maximum degree k+2 that are k-ordered for even k. A concept related to k-orderedness, namely that of k-edge-orderedness, is likewise strongly related to connectivity properties. We study this relation and give bounds on the connectivity necessary to imply k-(edge-)orderedness properties.     

The <Emphasis Type="Italic">k</Emphasis>-domatic number of a graph

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A k-dominating set of the graph G is a subset D of V (G) such that every vertex of V (G)-D is adjacent to at least k vertices in D. A k-domatic partition of G is a partition of V (G) into k-dominating sets. The maximum number of dominating sets in a k-domatic partition of G is called the k-domatic number d _k (G). In this paper, we present upper and lower bounds for the k-domatic number, and we establish Nordhaus-Gaddum-type results. Some of our results extend those for the classical domatic number d(G) = d ₁(G).      

6-path-connectivity and 6-generation

We consider simple connected graphs for which there is a path of length at least 6 between every pair of distinct vertices. We wish to show that in these graphs the cycle space over Z₂ is generated by the cycles of length at least 6. Furthermore, we wish to generalize the result for k-path-connected graphs which contain a long cycle.     

H-domination in graphs

Let H be some fixed graph of order p. For a given graph G and vertex set S⊆V(G), we say that S is H-decomposable if S can be partitioned as S=S₁∪S₂∪?∪S_j where, for each of the disjoint subsets S_i, with 1?i?j, we have |S_i|=p and H is a spanning subgraph of 〈S_i〉, the subgraph induced by S_i. We define the H-domination number of G, denoted as γ_H(G), to be the minimum cardinality of an H-decomposable dominating set S. If no such dominating set exists, we write γ_H(G)=∞. We show that the associated H-domination decision problem is NP-complete for every choice of H. Bounds are shown for γ_H(G). We show, in particular, that if δ(G)?2, then γ_P3(G)?3γ(G). Also, if γ_P3(G)=3γ(G), then every γ(G)-set is an efficient dominating set.     

A characterization of maximal non-k-factor-critical graphs

A graph G of order p is k-factor-critical,where p and k are positive integers with the same parity, if the deletion of any set of k vertices results in a graph with a perfect matching. G is called maximal non-k-factor-critical if G is not k-factor-critical but G+e is k-factor-critical for every missing edge e∉E(G). A connected graph G with a perfect matching on 2n vertices is k-extendable, for 1?k?n-1, if for every matching M of size k in G there is a perfect matching in G containing all edges of M. G is called maximal non-k-extendable if G is not k-extendable but G+e is k-extendable for every missing edge e∉E(G) . A connected bipartite graph G with a bipartitioning set (X,Y) such that |X|=|Y|=n is maximal non-k-extendable bipartite if G is not k-extendable but G+xy is k-extendable for any edge xy∉E(G) with x∈X and y∈Y. A complete characterization of maximal non-k-factor-critical graphs, maximal non-k-extendable graphs and maximal non-k-extendable bipartite graphs is given.     

Quartically hyponormal weighted shifts need not be 3-hyponormal

We give the first example of a quartically hyponormal unilateral weighted shift which is not 3-hyponormal.     

Extremal double hexagonal chains with respect to k-matchings and k-independent sets

“Double hexagonal chains” can be considered as benzenoids constructed by successive fusions of successive naphthalenes along a zig-zag sequence of triples of edges as appear on opposite sides of each naphthalene unit. In this paper, we discuss the numbers of k-matchings and k-independent sets of double hexagonal chains, as well as Hosoya indices and Merrifield-Simmons indices, and obtain some extremal results: among all the double hexagonal chains with the same number of naphthalene units, (a) the double linear hexagonal chain has minimal k-matching number and maximal k-independent set number and (b) the double zig-zag hexagonal chain has maximal k-matching number and minimal k-independent set number, which are extensions to hexagonal chains [L. Zhang and F. Zhang, Extremal hexagonal chains concerning k-matchings and k-independent sets, J. Math. Chem. 27 (2000) 319-329].     

On 3-regular 4-ordered graphs

A simple graph G is k-ordered (respectively, k-ordered hamiltonian), if for any sequence of k distinct vertices v₁,…,v_kof G there exists a cycle (respectively, hamiltonian cycle) in G containing these k vertices in the specified order. In 1997 Ng and Schultz introduced these concepts of cycle orderability and posed the question of the existence of 3-regular 4-ordered (hamiltonian) graphs other than K₄ and K_3,3. Ng and Schultz observed that a 3-regular 4-ordered graph on more than 4 vertices is triangle free. We prove that a 3-regular 4-ordered graph G on more than 6 vertices is square free,and we show that the smallest graph that is triangle and square free, namely the Petersen graph, is 4-ordered. Furthermore, we prove that the smallest graph after K₄ and K_3,3 that is 3-regular 4-ordered hamiltonianis the Heawood graph. Finally, we construct an infinite family of 3-regular 4-ordered graphs.     

Multidimensional bipartite trees

We introduce a bipartite version of k-trees and establish characterizations of bipartite 2- and 3-trees.     

Some properties and characterizations for generalized multivariate Pareto distributions

In this paper, several distributional properties and characterization theorems of the generalized multivariate Pareto distributions are studied. It is found that the multivariate Pareto distributions have many mixture properties. They are mixed either by geometric, Weibull, or exponential variables. The multivariate Pareto, MP^(k)(I), MP^(k)(II), and MP^(k)(IV) families have closure property under finite sample minima. The MP^(k)(III) family is closed under both geometric minima and geometric maxima. Through the geometric minima procedure, one characterization theorem for MP^(k)(III) distribution is developed. Moreover, the MP^(k)(III) distribution is proved as the limit multivariate distribution under repeated geometric minimization. Also, a characterization theorem for the homogeneous MP^(k)(IV) distribution via the weighted minima among the ordered coordinates is developed. Finally, the MP^(k)(II) family is shown to have the truncation invariant property.     

A note on k-potence preserving maps

Let M_n be the algebra of all n×n complex matrices and Γ_n the set of all k-potent matrices in M_n. Suppose ?:M_n→M_n is a map satisfying A-λB∈Γ_n implies ?(A)-λ?(B)∈Γ_n, where A, B∈M_n, λ∈C. Then either ? is of the form ?(A)=cTAT^-1, A∈M_n, or ? is of the form ?(A)=cTA^tT^-1, A∈M_n, where T∈M_n is an invertible matrix, c∈C satisfies c^k=c.     

Subclasses of k-trees: Characterization and recognition

A k-tree is either a complete graph on k vertices or a graph G=(V,E) that contains a vertex whose neighbourhood in G induces a complete graph on k vertices and whose removal results in a k-tree. We present two new subclasses of k-trees and their properties. First, we present the definition and characterization of k-path graphs, based on the concept of k-paths, that generalizes the classic concept of paths. We also introduce the simple-clique k-trees, of which the maximal outerplanar graphs and the planar 3-trees are particular cases. Based on Characterization Theorems, we show recognition algorithms for both families. Finally, we establish the inclusion relations among these new classes and k-trees.     

On extremal k-outconnected graphs

Let G be a minimally k-connected graph with n nodes and m edges. Mader proved that if n?3k-2 then m?k(n-k), and for n?3k-1 an equality is possible if, and only if, G is the complete bipartite graph K_k,n-k. Cai proved that if n?3k-2 then m?⌊(n+k)²/8⌋, and listed the cases when this bound is tight.In this paper we prove a more general theorem, which implies similar results for minimally k-outconnected graphs; a graph is called k-outconnected from r if it contains k internally disjoint paths from r to every other node.     

Extremal k-edge-hamiltonian hypergraphs

An r-uniform hypergraph is k-edge-hamiltonian iff it still contains a hamiltonian-chain after deleting any k edges of the hypergraph. What is the minimum number of edges in such a hypergraph? We give lower and upper bounds for this question for several values of r and k.     

A new family of k-Fibonacci numbers

In the present paper, we give a new family of k-Fibonacci numbers and establish some properties of the relation to the ordinary Fibonacci numbers. Furthermore, we describe the recurrence relations and the generating functions of the new family for k=2 and k=3, and presents a few identity formulas for the family and the ordinary Fibonacci numbers.     

Identifying large robust network clusters via new compact formulations of maximum k-club problems

Network robustness issues are crucial in a variety of application areas. In many situations, one of the key robustness requirements is the connectivity between each pair of nodes through a path that is short enough, which makes a network cluster more robust with respect to potential network component disruptions. A k-club, which by definition is a subgraph of a diameter of at most k, is a structure that addresses this requirement (assuming that k is small enough with respect to the size of the original network). We develop a new compact linear 0-1 programming formulation for finding maximum k-clubs that has substantially fewer entities compared to the previously known formulation (O(kn²) instead of O(n^k+1), which is important in the general case of k > 2) and is rather tight despite its compactness. Moreover, we introduce a new related concept referred to as an R-robust k-club (or, (k, R)-club), which naturally arises from the developed k-club formulations and extends the standard definition of a k-club by explicitly requiring that there must be at least R distinct paths of length at most k between all pairs of nodes. A compact formulation for the maximum R-robust k-club problem is also developed, and error and attack tolerance properties of the important special case of R-robust 2-clubs are investigated. Computational results are presented for multiple types of random graph instances.     

Fast block diagonalization of k-tridiagonal matrices

In the present paper, we give a fast algorithm for block diagonalization of k-tridiagonal matrices. The block diagonalization provides us with some useful results: e.g., another derivation of a very recent result on generalized k-Fibonacci numbers in [M.E.A. El-Mikkawy, T. Sogabe, A new family of k-Fibonacci numbers, Appl. Math. Comput. 215 (2010) 4456-4461]; efficient (symbolic) algorithm for computing the matrix determinant.