Exponents of 2-coloring of symmetric digraphs

A 2-coloring (G₁,G₂) of a digraph is 2-primitive if there exist nonnegative integers h and k with h+k>0 such that for each ordered pair (u,v) of vertices there exists an (h,k)-walk in (G₁,G₂) from u to v. The exponent of (G₁,G₂) is the minimum value of h+k taken over all such h and k. In this paper, we consider 2-colorings of strongly connected symmetric digraphs with loops, establish necessary and sufficient conditions for these to be 2-primitive and determine an upper bound on their exponents. We also characterize the 2-colored digraphs that attain the upper bound and the exponent set for this family of digraphs on n vertices.     

Bounds on the eigenvalues of graphs with cut vertices or edges

In this paper, we characterize the extremal graph having the maximal Laplacian spectral radius among the connected bipartite graphs with n vertices and k cut vertices, and describe the extremal graph having the minimal least eigenvalue of the adjacency matrices of all the connected graphs with n vertices and k cut edges. We also present lower bounds on the least eigenvalue in terms of the number of cut vertices or cut edges and upper bounds on the Laplacian spectral radius in terms of the number of cut vertices.     

Connected graphs with maximal Q-index: The one-dominating-vertex case

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. For every pair of positive integers n,k, it is proved that if 3?k?n-3, then H_n,k, the graph obtained from the star K_1,n-1 by joining a vertex of degree 1 to k+1 other vertices of degree 1, is the unique connected graph that maximizes the largest signless Laplacian eigenvalue over all connected graphs with n vertices and n+k edges.     

On Turan hypergraphs

Let α(H) be the stability number of a hypergraph H = (X, $\text{E}$). T(n, k, α) is the smallest q such that there exists a k-uniform hypergraph H with n vertices, q edges and with α(H) ? α. A k-uniform hypergraph H, with n vertices, T(n, k, α) edges and α(H) ?α is a Turan hypergraph. The value of T(n, 2, α) is given by a theorem of Turan. In this paper new lower bounds to T(n, k, α) are obtained and it is proved that an infinity of affine spaces are Turan hypergraphs.     

Graphs with given number of cut vertices and extremal Merrifield-Simmons index

The Merrifield-Simmons index of a graph is defined as the total number of its independent sets, including the empty set. Denote by G(n,k) the set of connected graphs with n vertices and k cut vertices. In this paper, we characterize the graphs with the maximum and minimum Merrifield-Simmons index, respectively, among all graphs in G(n,k) for all possible k values.     

Counting strongly‐connected,moderately sparse directed graphs

A sharp asymptotic formula for the number of strongly connected digraphs on n labelled vertices with m arcs, under the condition m ‐ n ? n^2/3, m = O(n), is obtained; this provides a partial solution of a problem posed by Wright back in 1977. This formula is a counterpart of a classic asymptotic formula, due to Bender, Canfield and McKay, for the total number of connected undirected graphs on n vertices with m edges. A key ingredient of their proof was a recurrence equation for the connected graphs count due to Wright. No analogue of Wright's recurrence seems to exist for digraphs. In a previous paper with Nick Wormald we rederived the BCM formula by counting first connected graphs among the graphs of minimum degree 2, at least. In this paper, using a similar embedding for directed graphs, we find an asymptotic formula, which includes an explicit error term, for the fraction of strongly‐connected digraphs with parameters m and n among all such digraphs with positive in/out‐degrees. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Broadcasting from multiple originators

We begin an investigation of broadcasting from multiple originators, a variant of broadcasting in which any k vertices may be the originators of a message in a network of n vertices. The requirement is that the message be distributed to all n vertices in minimum time. A minimumk-originator broadcast graph is a graph on n vertices with the fewest edges such that any subset of k vertices can broadcast in minimum time. B_k(n) is the number of edges in such a graph. In this paper, we present asymptotic upper and lower bounds on B_k(n). We also present an exact result for the case when . We also give an upper bound on the number of edges in a relaxed version of this problem in which one additional time unit is allowed for the broadcast.     

On pancyclic digraphs

We show that a strongly connected digraph with n vertices and minimum degree ? n is pancyclic unless it is one of the graphs K_p,p. This generalizes a result of A. Ghouila-Houri. We disprove a conjecture of J. A. Bondy by showing that there exist hamiltonian digraphs with n vertices and $\text{1}\text{2}\text{n(n + 1) – 3}$ edges which are not pancyclic. We show that any hamiltonian digraph with n vertices and at least $\text{1}\text{2}\text{n(n + 1) – 1}$ edges is pancyclic and we give some generalizations of this result. As applications of these results we determine the minimal number of edges required in a digraph to guarantee the existence of a cycle of length k, k ? 2, and we consider the corresponding problem where the digraphs under consideration are assumed to be strongly connected.     

The Laplacian spectral radius of some bipartite graphs

In this paper, we study the largest Laplacian spectral radius of the bipartite graphs with n vertices and k cut edges and the bicyclic bipartite graphs, respectively. Identifying the center of a star K_1,k and one vertex of degree n of K_m,n, we denote by the resulting graph. We show that the graph (1?k?n-4) is the unique graph with the largest Laplacian spectral radius among the bipartite graphs with n vertices and k cut edges, and (n?7) is the unique graph with the largest Laplacian spectral radius among all the bicyclic bipartite graphs.     

An interpretation of the Dittert conjecture in terms of semi-matchings

Let G be K_n,n with non-negative edge weights and let U and V be the two colour classes of vertices in G. We define a k-semimatching in G to be a set of k edges such that the edges either have distinct ends in U or distinct ends in V. Semimatchings are to be counted according to the product of the weights on the edges in the semimatching. The Dittert conjecture is a longstanding open problem involving matrix permanents. Here we show that it is equivalent to the following assertion: For a fixed total weight, the number of n-semimatchings in G is maximised by weighting all edges of G equally. We also introduce sub-Dittert functions which count k-semimatchings and are analogous to the subpermanent functions which count k-matchings. We prove some results about the extremal values of our sub-Dittert functions, and also that the Dittert conjecture cannot be disproved by means of unweighted graphs.     

On the signless Laplacian spectral radius of graphs with cut vertices

In this paper, we show that among all the connected graphs with n vertices and k cut vertices, the maximal signless Laplacian spectral radius is attained uniquely at the graph G_n,k, where G_n,k is obtained from the complete graph K_n-k by attaching paths of almost equal lengths to all vertices of K_n-k. We also give a new proof of the analogous result for the spectral radius of the connected graphs with n vertices and k cut vertices (see [A. Berman, X.-D. Zhang, On the spectral radius of graphs with cut vertices, J. Combin. Theory Ser. B 83 (2001) 233-240]). Finally, we discuss the limit point of the maximal signless Laplacian spectral radius.     

An extremal problem in hypergraph coloring

F(n, r, k) is defined to be the minimum number of edges in an r graph on n vertices in which there exists a strongly colored edge in every equipartite k-coloring. Approximate values are given for this function in the general case, and the problem is solved in the particular case of graphs.     

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.     

Perfect matchings in large uniform hypergraphs with large minimum collective degree

We define a perfect matching in a k-uniform hypergraph H on n vertices as a set of ⌊n/k⌋ disjoint edges. Let δk−1(H) be the largest integer d such that every (k−1)-element set of vertices of H belongs to at least d edges of H.In this paper we study the relation between δk−1(H) and the presence of a perfect matching in H for k?3. Let t(k,n) be the smallest integer t such that every k-uniform hypergraph on n vertices and with δk−1(H)?t contains a perfect matching.For large n divisible by k, we completely determine the values of t(k,n), which turn out to be very close to n/2−k. For example, if k is odd and n is large and even, then t(k,n)=n/2−k+2. In contrast, for n not divisible by k, we show that t(k,n)∼n/k.In the proofs we employ a newly developed “absorbing” technique, which has a potential to be applicable in a more general context of establishing existence of spanning subgraphs of graphs and hypergraphs.     

The diameter of the acyclic Birkhoff polytope

In this work we give an interpretation of vertices and edges of the acyclic Birkhoff polytope, T_n=Ω_n(T), where T is a tree with n vertices, in terms of graph theory. We generalize a recent result relatively to the diameter of the graph G(T_n).     

Bijective linear maps on semimodules spanned by Boolean matrices of fixed rank

Let M_m,n(B) be the semimodule of all m×n Boolean matrices where B is the Boolean algebra with two elements. Let k be a positive integer such that 2?k?min(m,n). Let B(m,n,k) denote the subsemimodule of M_m,n(B) spanned by the set of all rank k matrices. We show that if T is a bijective linear mapping on B(m,n,k), then there exist permutation matrices P and Q such that T(A)=PAQ for all A∈B(m,n,k) or m=n and T(A)=PA^tQ for all A∈B(m,n,k). This result follows from a more general theorem we prove concerning the structure of linear mappings on B(m,n,k) that preserve both the weight of each matrix and rank one matrices of weight k². Here the weight of a Boolean matrix is the number of its nonzero entries.     

The signless Laplacian spectral radius of tricyclic graphs and trees with k pendant vertices

In this paper, we consider the following problem: of all tricyclic graphs or trees of order n with k pendant vertices (n,k fixed), which achieves the maximal signless Laplacian spectral radius?We determine the graph with the largest signless Laplacian spectral radius among all tricyclic graphs with n vertices and k pendant vertices. Then we show that the maximal signless Laplacian spectral radius among all trees of order n with k pendant vertices is obtained uniquely at T_n,k, where T_n,k is a tree obtained from a star K_1,k and k paths of almost equal lengths by joining each pendant vertex to one end-vertex of one path. We also discuss the signless Laplacian spectral radius of T_n,k and give some results.     

Extremal problems on detectable colorings of trees

Let G be a connected graph of order 3 or more and let be a coloring of the edges of G (where adjacent edges may be colored the same). For each vertex v of G, the color code of v is the k-tuple c(v)=(a₁,a₂,…,a_k), where a_i is the number of edges incident with v that are colored i (1?i?k). The coloring c is called detectable if distinct vertices have distinct color codes; while the detection number det(G) of G is the minimum positive integer k for which G has a detectable k-coloring. For each integer n?3, let D_T(n) be the maximum detection number among all trees of order n and d_T(n) the minimum detection number among all trees of order n. The numbers D_T(n) and d_T(n) are determined for all integers n?3. Furthermore, it is shown that for integers k?2 and n?3, there exists a tree T of order n having det(T)=k if and only if d_T(n)?k?D_T(n).     

Lower bound of the hadwiger number of graphs by their average degree

The aim of this paper is to show that the minimum Hadwiger number of graphs with average degreek isO(k/√logk). Specially, it follows that Hadwiger’s conjecture is true for almost all graphs withn vertices, furthermore ifk is large enough then for almost all graphs withn vertices andnk edges.     

Extremal edge problems for graphs with given hyperoctahedral automorphism group

For k > 1, let H_k denote the hyperoctahedral group S_k[S₂] of order 2^kk!. An (H_k, n)- graph is a graph on n vertices with automorphism group abstractly isomorphic to H_k. For each k an (H_k, n)-graph exists precisely when n ? 2k; for each n ? 2k the minimum and maximum number of edges possible for such graphs are determined. The analogous results for connected (H_k, n)-graphs are also obtained.