Edge disjoint spanning trees in random graphs

We show that almost everyG _m-out containsm edge disjoint spanning trees.     

Bounds on the convex label number of trees

A convex labelling of a tree is an assignment of distinct non-negative integer labels to vertices such that wheneverx, y andz are the labels of vertices on a path of length 2 theny≦(x+z)/2. In addition if the tree is rooted, a convex labelling must assign 0 to the root. The convex label number of a treeT is the smallest integerm such thatT has a convex labelling with no label greater thanm. We prove that every rooted tree (and hence every tree) withn vertices has convex label number less than 4n. We also exhibitn-vertex trees with convex label number 4n/3+o(n), andn-vertex rooted trees with convex label number 2n +o(n). The research by M. B. and A. W. was partly supported by NSF grant MCS—8311422.     

The number of spanning trees of a graph

In this paper, we present some sharp upper bounds for the number of spanning trees of a connected graph in terms of its structural parameters such as the number of vertices, the number of edges, maximum vertex degree, minimum vertex degree, connectivity and chromatic number.     

The number of spanning trees in regular graphs

Let C(G) denote the number of spanning trees of a graph G. It is shown that there is a function ?(k) that tends to zero as k tends to infinity such that for every connected, k-regular simple graph G on n vertices C(G) = {k[1 ? δ(G)]}ⁿ. where 0 ≤ δ(G) ≤ ?(k).     

Minimal spanning trees with a constraint on the number of leaves

In this paper we discuss minimal spanning trees with a constraint on the number of leaves. Tree topologies appear when designing centralized terminal networks. The constraint on the number of leaves arises because the software and hardware associated to each terminal differs accordingly with its position in the tree. Usually, the software and hardware associated to a “degree-1” terminal is cheaper than the software and hardware used in the remaining terminals because for any intermediate terminal j one needs to check if the arrival message is destined to that node or to any other node located after node j. As a consequence, that particular terminal needs software and hardware for message routing. On the other hand, such equipment is not needed in “degree-1” terminals. Assuming that the hardware and software for message routing in the nodes is already available, the above discussion motivates a constraint stating that a tree solution has to contain exactly a certain number of “degree-1” terminals. We present two different formulations for this problem and some lower bounding schemes derived from them. We discuss a simple local-exchange heuristic and present computational results taken from a set of complete graphs with up to 40 nodes. Integer Linear Programming formulations for related problems are also discussed at the end.     

Interpolation theorem for the number of end-vertices of spanning trees

The following interpolation theorem is proved: If a graph G contains spanning trees having exactly m and n end-vertices, with m < n, then for every integer k, m < k < n, G contains a spanning tree having exactly k end-vertices. This settles a problem posed by Chartrand at the Fourth International Conference on Graph Theory and Applications held in Kalamazoo, 1980.     

A lower bound on the number of spanning trees with k end-vertices

If a graph G with cycle rank ρ contains both spanning trees with m and with n end-vertices, m < n, then G has at least 2ρ spanning trees with k end-vertices for each integer k, m < k < n. Moreover, the lower bound of 2ρ is best possible.     

Connected graphs with a minimal number of spanning trees

The problem studied is the following: Find a simple connected graph G with given numbers of vertices and edges which minimizes the number t_μ(G), the number of spanning trees of the multigraph obtained from G by adding μ parallel edges between every pair of distinct vertices. If G is nearly complete (the number of edges qis ≥(₂^P)?p+2 where p is the number of vertices), then the solution to the minimization problem is unique (up to isomorphism) and the same for all values of μ. The present paper investigates the case whereq<(₂^P)?p+2. In this case the solution is not always unique and there does not always exist a common solution for all values of μ. A (small) class of graphs is given such that for any μ there exists a solution to the problem which is contained in this class. For μ = 0 there is only one graph in the class which solves the problem. This graph is described and the minimum value of t₀(G) is found. In order to derive these results a representation theorem is proved for the cofactors of a special class of matrices which contains the tree matrices associated with graphs.     

The asymptotic number of spanning trees in circulant graphs

Let T(G) be the number of spanning trees in graph G. In this note, we explore the asymptotics of T(G) when G is a circulant graph with given jumps.The circulant graph is the 2k-regular graph with n vertices labeled 0,1,2,…,n−1, where node i has the 2k neighbors i±s₁,i±s₂,…,i±s_k where all the operations are . We give a closed formula for the asymptotic limit as a function of s₁,s₂,…,s_k. We then extend this by permitting some of the jumps to be linear functions of n, i.e., letting s_i, d_i and e_i be arbitrary integers, and examining

     

Further analysis of the number of spanning trees in circulant graphs

Let 1?s₁<s₂<?<s_k?⌊n/2⌋ be given integers. An undirected even-valent circulant graph, has n vertices 0,1,2,…, n-1, and for each and j(0?j?n-1) there is an edge between j and . Let stand for the number of spanning trees of . For this special class of graphs, a general and most recent result, which is obtained in [Y.P. Zhang, X. Yong, M. Golin, [The number of spanning trees in circulant graphs, Discrete Math. 223 (2000) 337-350]], is that where a_n satisfies a linear recurrence relation of order 2^s_k-1. And, most recently, for odd-valent circulant graphs, a nice investigation on the number a_n is [X. Chen, Q. Lin, F. Zhang, The number of spanning trees in odd-valent circulant graphs, Discrete Math. 282 (2004) 69-79].In this paper, we explore further properties of the numbers a_n from their combinatorial structures. Comparing with the previous work, the differences are that (1) in finding the coefficients of recurrence formulas for a_n, we avoid solving a system of linear equations with exponential size, but instead, we give explicit formulas; (2) we find the asymptotic functions and therefore we ‘answer’ the open problem posed in the conclusion of [Y.P. Zhang, X. Yong, M. Golin, The number of spanning trees in circulant graphs, Discrete Math. 223 (2000) 337-350]. As examples, we describe our technique and the asymptotics of the numbers.     

Bounds for the disjoint unions theorem

We give a short self-contained proof of the disjoint unions theorem of Graham and Rothschild and of the non-repeating sums theorem of Rado, Folkman, and Sanders. The proof yields an iterated exponential upper bound for the functions involved in these results.     

On the number of spanning trees in graphs with multiple edges

In this paper, we consider a class of nonlinear matrix equation of the type \(X+\sum _{i=1}^mA_i^{*}X^{-q}A_i-\sum _{j=1}^nB_{j}^{*}X^{-r}B_j=Q\), where \(0<q,\,r\le 1\) and Q is positive definite. Based on the Schauder fixed point theorem and Bhaskar–Lakshmikantham coupled fixed point theorem, we derive some sufficient conditions for the existence and uniqueness of the positive definite solution to such equations. An iterative method is provided to compute the unique positive definite solution. A perturbation estimation and the explicit expression of Rice condition number of the unique positive definite solution are also established. The theoretical results are illustrated by numerical examples.     

On the number of rainbow spanning trees in edge-colored complete graphs

A spanning tree of a properly edge-colored complete graph, $K_n$, is rainbow provided that each of its edges receives a distinct color. In 1996, Brualdi and Hollingsworth conjectured that if $K_{2m}$ is properly $(2m?1)$-edge-colored, then the edges of $K_{2m}$ can be partitioned into $m$ rainbow spanning trees except when $m=2$. By means of an explicit, constructive approach, in this paper we construct $?\sqrt{6m+9}∕3?$ mutually edge-disjoint rainbow spanning trees for any positive value of $m$. Not only are the rainbow trees produced, but also some structure of each rainbow spanning tree is determined in the process. This improves upon best constructive result to date in the literature which produces exactly three rainbow trees.     

Undirected simple connected graphs with minimum number of spanning trees

We show that for positive integers n, m with n(n−1)/2≥m≥n−1, the graph L_n,m having n vertices and m edges that consists of an (n−k)-clique and k−1 vertices of degree 1 has the fewest spanning trees among all connected graphs on n vertices and m edges. This proves Boesch’s conjecture [F.T. Boesch, A. Satyanarayana, C.L. Suffel, Least reliable networks and reliability domination, IEEE Trans. Commun. 38 (1990) 2004-2009].