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121.
A set S of vertices of a graph G=(V,E) is a dominating set if every vertex of V(G)?S is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G. The domination subdivision number is the minimum number of edges that must be subdivided in order to increase the domination number. Velammal showed that for any tree T of order at least 3, . In this paper, we give two characterizations of trees whose domination subdivision number is 3 and a linear algorithm for recognizing them. 相似文献
122.
Let p be a graph parameter that assigns a positive integer value to every graph. The inverse problem for p asks for a graph within a prescribed class (here, we will only be concerned with trees), given the value of p. In this context, it is of interest to know whether such a graph can be found for all or at least almost all integer values of p. We will provide a very general setting for this type of problem over the set of all trees, describe some simple examples and finally consider the interesting parameter “number of subtrees”, where the problem can be reduced to some number-theoretic considerations. Specifically, we will prove that every positive integer, with only 34 exceptions, is the number of subtrees of some tree. 相似文献
123.
We consider the effects on the algebraic connectivity of various graphs when vertices and graphs are appended to the original graph. We begin by considering weighted trees and appending a single isolated vertex to it by adding an edge from the isolated vertex to some vertex in the tree. We then determine the possible set vertices in the tree that can yield the maximum change in algebraic connectivity under such an operation. We then discuss the changes in algebraic connectivity of a star when various graphs such as trees and complete graphs are appended to its pendant vertices. 相似文献
124.
Charles R. Johnson Christopher Jordan-Squire David A. Sher 《Discrete Applied Mathematics》2010,158(6):681-691
Among the possible multiplicity lists for the eigenvalues of Hermitian matrices whose graph is a tree we focus upon M2, the maximum value of the sum of the two largest multiplicities. The corresponding M1 is already understood. The notion of assignment (of eigenvalues to subtrees) is formalized and applied. Using these ideas, simple upper and lower bounds are given for M2 (in terms of simple graph theoretic parameters), cases of equality are indicated, and a combinatorial algorithm is given to compute M2 precisely. In the process, several techniques are developed that likely have more general uses. 相似文献
125.
The atom-bond connectivity (ABC) index of a graph G is defined as
126.
Stavros D. Nikolopoulos 《Discrete Applied Mathematics》2002,120(1-3):165-195
A coloring of a graph G is an assignment of colors to its vertices so that no two adjacent vertices have the same color. We study the problem of coloring permutation graphs using certain properties of the lattice representation of a permutation and relationships between permutations, directed acyclic graphs and rooted trees having specific key properties. We propose an efficient parallel algorithm which colors an n-node permutation graph in O(log2 n) time using O(n2/log n) processors on the CREW PRAM model. Specifically, given a permutation π we construct a tree T*[π], which we call coloring-permutation tree, using certain combinatorial properties of π. We show that the problem of coloring a permutation graph is equivalent to finding vertex levels in the coloring-permutation tree. 相似文献
127.
Stavros D.?NikolopoulosEmail author Charis?Papadopoulos 《Graphs and Combinatorics》2004,20(3):383-397
In this paper we examine the classes of graphs whose Kn-complements are trees or quasi-threshold graphs and derive formulas for their number of spanning trees; for a subgraph H of Kn, the Kn-complement of H is the graph Kn–H which is obtained from Kn by removing the edges of H. Our proofs are based on the complement spanning-tree matrix theorem, which expresses the number of spanning trees of a graph as a function of the determinant of a matrix that can be easily constructed from the adjacency relation of the graph. Our results generalize previous results and extend the family of graphs of the form Kn–H admitting formulas for the number of their spanning trees.Final version received: March 18, 2004 相似文献
128.
We study the behaviour of harmonic functions on a homogeneous tree from the point of view of the tangential boundary covergence. 相似文献
129.
130.