Edge-connectivity and edge-disjoint spanning trees

Given a graph G, for an integer c∈{2,…,|V(G)|}, define λ_c(G)=min{|X|:X⊆E(G),ω(G−X)≥c}. For a graph G and for an integer c=1,2,…,|V(G)|−1, define,

     

Independence polynomials of k-tree related graphs

An independent set of a graph G is a set of pairwise non-adjacent vertices. Let α(G) denote the cardinality of a maximum independent set and f_s(G) for 0≤s≤α(G) denote the number of independent sets of s vertices. The independence polynomial defined first by Gutman and Harary has been the focus of considerable research recently. Wingard bounded the coefficients f_s(T) for trees T with n vertices: for s≥2. We generalize this result to bounds for a very large class of graphs, maximal k-degenerate graphs, a class which includes all k-trees. Additionally, we characterize all instances where our bounds are achieved, and determine exactly the independence polynomials of several classes of k-tree related graphs. Our main theorems generalize several related results known before.     

Turán's theorem for pseudo-random graphs

The generalized Turán number ex(G,H) of two graphs G and H is the maximum number of edges in a subgraph of G not containing H. When G is the complete graph Km on m vertices, the value of ex(Km,H) is , where o(1)→0 as m→∞, by the Erd?s-Stone-Simonovits theorem.In this paper we give an analogous result for triangle-free graphs H and pseudo-random graphs G. Our concept of pseudo-randomness is inspired by the jumbled graphs introduced by Thomason [A. Thomason, Pseudorandom graphs, in: Random Graphs '85, Poznań, 1985, North-Holland, Amsterdam, 1987, pp. 307-331. MR 89d:05158]. A graph G is (q,β)-bi-jumbled if

     

Boundedness of singular integral operators with variable kernels

Let K be a generalized Calderón-Zygmund kernel defined on Rn×(Rn?{0}). The singular integral operator with variable kernel given by

     

The multiple originator broadcasting problem in graphs

Given a graph G and a vertex subset S of V(G), the broadcasting time with respect toS, denoted by b(G,S), is the minimum broadcasting time when using S as the broadcasting set. And the k-broadcasting number, denoted by b_k(G), is defined by b_k(G)=min{b(G,S)|S⊆V(G),|S|=k}.Given a graph G and two vertex subsets S, S^′ of V(G), define , d(S,S^′)=min{d(u,v)|u∈S, v∈S^′}, and for all v∈V(G). For all k, 1?k?|V(G)|, the k-radius of G, denoted by r_k(G), is defined as r_k(G)=min{d(G,S)|S⊆V(G), |S|=k}.In this paper, we study the relation between the k-radius and the k-broadcasting numbers of graphs. We also give the 2-radius and the 2-broadcasting numbers of the grid graphs, and the k-broadcasting numbers of the complete n-partite graphs and the hypercubes.     

Linkability in iterated line graphs

We prove that for every graph H with the minimum degree δ?5, the third iterated line graph L³(H) of H contains as a minor. Using this fact we prove that if G is a connected graph distinct from a path, then there is a number k_G such that for every i?k_G the i-iterated line graph of G is -linked. Since the degree of Lⁱ(G) is even, the result is best possible.     

Decomposition rank and absorbing extensions of type I algebras

We prove the following: Let A and B be separable C^*-algebras. Suppose that B is a type I C^*-algebra such that

(i)

B has only infinite dimensional irreducible *-representations, and

(ii)

B has finite decomposition rank.

If

0→B→C→A→0     

On multicolour noncomplete Ramsey graphs of star graphs

Given graphs , where k≥2, the notation

     

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.     

On the signed edge domination number of graphs

Let be the signed edge domination number of G. In 2006, Xu conjectured that: for any 2-connected graph G of order n(n≥2), . In this article we show that this conjecture is not true. More precisely, we show that for any positive integer m, there exists an m-connected graph G such that . Also for every two natural numbers m and n, we determine , where K_m,n is the complete bipartite graph with part sizes m and n.     

On the minimum real roots of the adjoint polynomial of a graph

For a graph G, we denote by h(G,x) the adjoint polynomial of G. Let β(G) denote the minimum real root of h(G,x). In this paper, we characterize all the connected graphs G with .     

A note on path kernels and partitions

The detour order of a graph G, denoted by τ(G), is the order of a longest path in G. A subset S of V(G) is called a P_n-kernel of G if τ(G[S])≤n−1 and every vertex v∈V(G)−S is adjacent to an end-vertex of a path of order n−1 in G[S]. A partition of the vertex set of G into two sets, A and B, such that τ(G[A])≤a and τ(G[B])≤b is called an (a,b)-partition of G. In this paper we show that any graph with girth g has a P_n+1-kernel for every . Furthermore, if τ(G)=a+b, 1≤a≤b, and G has girth greater than , then G has an (a,b)-partition.     

A sufficient condition for pancyclability of graphs

Let G be a graph of order n and S be a vertex set of q vertices. We call G,S-pancyclable, if for every integer i with 3≤i≤q there exists a cycle C in G such that |V(C)∩S|=i. For any two nonadjacent vertices u,v of S, we say that u,v are of distance two in S, denoted by d_S(u,v)=2, if there is a path P in G connecting u and v such that |V(P)∩S|≤3. In this paper, we will prove that if G is 2-connected and for all pairs of vertices u,v of S with d_S(u,v)=2, , then there is a cycle in G containing all the vertices of S. Furthermore, if for all pairs of vertices u,v of S with d_S(u,v)=2, , then G is S-pancyclable unless the subgraph induced by S is in a class of special graphs. This generalizes a result of Fan [G. Fan, New sufficient conditions for cycles in graphs, J. Combin. Theory B 37 (1984) 221-227] for the case when S=V(G).     

Riesz means associated with certain product type convex domain

In this paper we study the maximal operators and the convolution operators Tδ associated with multipliers of the form

     

Colouring games on outerplanar graphs and trees

Let f be a graph function which assigns to each graph H a non-negative integer f(H)≤|V(H)|. The f-game chromatic number of a graph G is defined through a two-person game. Let X be a set of colours. Two players, Alice and Bob, take turns colouring the vertices of G with colours from X. A partial colouring c of G is legal (with respect to graph function f) if for any subgraph H of G, the sum of the number of colours used in H and the number of uncoloured vertices of H is at least f(H). Both Alice and Bob must colour legally (i.e., the partial colouring produced needs to be legal). The game ends if either all the vertices are coloured or there are uncoloured vertices with no legal colour. In the former case, Alice wins the game. In the latter case, Bob wins the game. The f-game chromatic number of G, χ_g(f,G), is the least number of colours that the colour set X needs to contain so that Alice has a winning strategy. Let be the graph function defined as , for any n≥3 and otherwise. Then is called the acyclic game chromatic number of G. In this paper, we prove that any outerplanar graph G has acyclic game chromatic number at most 7. For any integer k, let ?_k be the graph function defined as ?_k(K₂)=2 and ?_k(P_k)=3 (P_k is the path on k vertices) and ?_k(H)=0 otherwise. This paper proves that if k≥8 then for any tree T, χ_g(?_k,T)≤9. On the other hand, if k≤6, then for any integer n, there is a tree T such that χ_g(?_k,T)≥n.     

Numerical radius and zero pattern of matrices

Let A be an n×n complex matrix and r be the maximum size of its principal submatrices with no off-diagonal zero entries. Suppose A has zero main diagonal and x is a unit n-vector. Then, letting ‖A‖ be the Frobenius norm of A, we show that

|〈Ax,x〉^|2?(1−1/2r−1/2n)‖A^‖2.     

Strong characterizing sequences of countable groups

András Biró and Vera Sós prove that for any subgroup G of T generated freely by finitely many generators there is a sequence A⊆N such that for all β∈T we have (‖.‖ denotes the distance to the nearest integer)

     

Graphic sequences with a realization containing a complete multipartite subgraph

A nonincreasing sequence of nonnegative integers π=(d₁,d₂,…,d_n) is graphic if there is a (simple) graph G of order n having degree sequence π. In this case, G is said to realizeπ. For a given graph H, a graphic sequence π is potentiallyH-graphic if there is some realization of π containing H as a (weak) subgraph. Let σ(π) denote the sum of the terms of π. For a graph H and n∈Z⁺, σ(H,n) is defined as the smallest even integer m so that every n-term graphic sequence π with σ(π)≥m is potentially H-graphic. Let denote the complete t partite graph such that each partite set has exactly s vertices. We show that and obtain the exact value of σ(K_j+K_s,s,n) for n sufficiently large. Consequently, we obtain the exact value of for n sufficiently large.     

Distance-two labellings of Hamming graphs

Let j≥k≥0 be integers. An ?-L(j,k)-labelling of a graph G=(V,E) is a mapping ?:V→{0,1,2,…,?} such that |?(u)−?(v)|≥j if u,v are adjacent and |?(u)−?(v)|≥k if they are distance two apart. Let λ_j,k(G) be the smallest integer ? such that G admits an ?-L(j,k)-labelling. Define to be the smallest ? if G admits an ?-L(j,k)-labelling with ?(V)={0,1,2,…,?} and ∞ otherwise. An ?-cyclic L(j,k)-labelling is a mapping ?:V→Z_? such that |?(u)−?(v)|_?≥j if u,v are adjacent and |?(u)−?(v)|_?≥k if they are distance two apart, where |x|_?=min{x,?−x} for x between 0 and ?. Let σ_j,k(G) be the smallest ?−1 of such a labelling, and define similarly to . We determine λ_2,0, , σ_2,0 and for all Hamming graphs K_q1□K_q2□?□K_qd (d≥2, q₁≥q₂≥?≥q_d≥2) and give optimal labellings, with the only exception being for q≥4. We also prove the following “sandwich theorem”: If q₁ is sufficiently large then for any graph G between K_q1□K_q2 and K_q1□K_q2□?□K_qd, and moreover we give a labelling which is optimal for these eight invariants simultaneously.