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1.
For a countable ultrahomogeneous graph G=〈G,ρ〉G=G,ρ let P(G)P(G) denote the collection of sets A⊂GAG such that 〈A,ρ∩[A]2〉≅GA,ρ[A]2G. The order types of maximal chains in the poset 〈P(G)∪{∅},⊂〉P(G){}, are characterized as:  相似文献   

2.
Let K be a proper (i.e., closed, pointed, full convex) cone in Rn. An n×n matrix A is said to be K-primitive if there exists a positive integer k such that ; the least such k is referred to as the exponent of A and is denoted by γ(A). For a polyhedral cone K, the maximum value of γ(A), taken over all K-primitive matrices A, is called the exponent of K and is denoted by γ(K). It is proved that if K is an n-dimensional polyhedral cone with m extreme rays then for any K-primitive matrix A, γ(A)?(mA−1)(m−1)+1, where mA denotes the degree of the minimal polynomial of A, and the equality holds only if the digraph (E,P(A,K)) associated with A (as a cone-preserving map) is equal to the unique (up to isomorphism) usual digraph associated with an m×m primitive matrix whose exponent attains Wielandt's classical sharp bound. As a consequence, for any n-dimensional polyhedral cone K with m extreme rays, γ(K)?(n−1)(m−1)+1. Our work answers in the affirmative a conjecture posed by Steve Kirkland about an upper bound of γ(K) for a polyhedral cone K with a given number of extreme rays.  相似文献   

3.
The energy of a graph is equal to the sum of the absolute values of its eigenvalues. The energy of a matrix is equal to the sum of its singular values. We establish relations between the energy of the line graph of a graph G and the energies associated with the Laplacian and signless Laplacian matrices of G.  相似文献   

4.
An n × n sign pattern Sn is potentially nilpotent if there is a real matrix having sign pattern Sn and characteristic polynomial xn. A new family of sign patterns Cn with a cycle of every even length is introduced and shown to be potentially nilpotent by explicitly determining the entries of a nilpotent matrix with sign pattern Cn. These nilpotent matrices are used together with a Jacobian argument to show that Cn is spectrally arbitrary, i.e., there is a real matrix having sign pattern Cn and characteristic polynomial for any real μi. Some results and a conjecture on minimality of these spectrally arbitrary sign patterns are given.  相似文献   

5.
It is known that the max-algebraic powers Ar of a nonnegative irreducible matrix are ultimately periodic. This leads to the concept of attraction cone Attr(A, t), by which we mean the solution set of a two-sided system λt(A)Arx=Ar+tx, where r is any integer after the periodicity transient T(A) and λ(A) is the maximum cycle geometric mean of A. A question which this paper answers, is how to describe Attr(A,t) by a concise system of equations without knowing T(A). This study requires knowledge of certain structures and symmetries of periodic max-algebraic powers, which are also described. We also consider extremals of attraction cones in a special case, and address the complexity of computing the coefficients of the system which describes attraction cone.  相似文献   

6.
In this paper, we characterize all extremal trees with the largest Laplacian spectral radius in the set of all trees with a given degree sequence. Consequently, we also obtain all extremal trees with the largest Laplacian spectral radius in the sets of all trees of order n with the largest degree, the leaves number and the matching number, respectively.  相似文献   

7.
Motivated by a Mohar’s paper proposing “how to order trees by the Laplacian coefficients”, we investigate a partial ordering of trees with diameters 3 and 4 by the Laplacian coefficients. These results are used to determine several orderings of trees by the Laplacian coefficients.  相似文献   

8.
Distance-based methods such as UPGMA (Unweighted Pair Group Method with Arithmetic Mean) continue to play a significant role in phylogenetic research. We use polyhedral combinatorics to analyze the natural subdivision of the positive orthant induced by classifying the input vectors according to tree topologies returned by the algorithm. The partition lattice informs the study of UPGMA trees. We give a closed form for the extreme rays of UPGMA cones on n taxa, and compute the spherical volumes of the UPGMA cones for small n.  相似文献   

9.
The spectra of some trees and bounds for the largest eigenvalue of any tree   总被引:2,自引:0,他引:2  
Let T be an unweighted tree of k levels such that in each level the vertices have equal degree. Let nkj+1 and dkj+1 be the number of vertices and the degree of them in the level j. We find the eigenvalues of the adjacency matrix and Laplacian matrix of T for the case of two vertices in level 1 (nk = 2), including results concerning to their multiplicity. They are the eigenvalues of leading principal submatrices of nonnegative symmetric tridiagonal matrices of order k × k. The codiagonal entries for these matrices are , 2 ? j ? k, while the diagonal entries are 0, …, 0, ±1, in the case of the adjacency matrix, and d1d2, …, dk−1dk ± 1, in the case of the Laplacian matrix. Finally, we use these results to find improved upper bounds for the largest eigenvalue of the adjacency matrix and of the Laplacian matrix of any given tree.  相似文献   

10.
Let G=(V(G),E(G)) be a unicyclic simple undirected graph with largest vertex degree Δ. Let Cr be the unique cycle of G. The graph G-E(Cr) is a forest of r rooted trees T1,T2,…,Tr with root vertices v1,v2,…,vr, respectively. Let
  相似文献   

11.
We provide positive answers to some open questions presented recently by Kim and Shader on a continuity-like property of the P-vertices of nonsingular matrices whose graph is a path. A criterion for matrices associated with more general trees to have at most n − 1 P-vertices is established. The cases of the cycles and stars are also analyzed. Several algorithms for generating matrices with a given number of P-vertices are proposed.  相似文献   

12.
A Bethe tree Bd,k is a rooted unweighted of k levels in which the root vertex has degree equal to d, the vertices at level j(2?j?k-1) have degree equal to (d+1) and the vertices at level k are the pendant vertices. In this paper, we first derive an explicit formula for the eigenvalues of the adjacency matrix of Bd,k. Moreover, we give the corresponding multiplicities. Next, we derive an explicit formula for the simple nonzero eigenvalues, among them the largest eigenvalue, of the Laplacian matrix of Bd,k. Finally, we obtain upper bounds on the largest eigenvalue of the adjacency matrix and of the Laplacian matrix of any tree T. These upper bounds are given in terms of the largest vertex degree and the radius of T, and they are attained if and only if T is a Bethe tree.  相似文献   

13.
For positive integers k and m, and a digraph D, the k-step m-competition graph of D has the same set of vertices as D and an edge between vertices x and y if and only if there are distinct m vertices v1,v2,…,vm in D such that there are directed walks of length k from x to vi and from y to vi for 1?i?m. In this paper, we present the definition of m-competition index for a primitive digraph. The m-competition index of a primitive digraph D is the smallest positive integer k such that is a complete graph. We study m-competition indices of primitive digraphs and provide an upper bound for the m-competition index of a primitive digraph.  相似文献   

14.
A generalized Bethe tree is a rooted unweighted tree in which vertices at the same level have the same degree. Let B be a generalized Bethe tree. The algebraic connectivity of:
the generalized Bethe tree B,
a tree obtained from the union of B and a tree T isomorphic to a subtree of B such that the root vertex of T is the root vertex of B,
a tree obtained from the union of r generalized Bethe trees joined at their respective root vertices,
a graph obtained from the cycle Cr by attaching B, by its root, to each vertex of the cycle, and
a tree obtained from the path Pr by attaching B, by its root, to each vertex of the path,
is the smallest eigenvalue of a special type of symmetric tridiagonal matrices. In this paper, we first derive a procedure to compute a tight upper bound on the smallest eigenvalue of this special type of matrices. Finally, we apply the procedure to obtain a tight upper bound on the algebraic connectivity of the above mentioned graphs.
  相似文献   

15.
For a positive integer m where 1?m?n, the m-competition index (generalized competition index) of a primitive digraph is the smallest positive integer k such that for every pair of vertices x and y, there exist m distinct vertices v1,v2,…,vm such that there are directed walks of length k from x to vi and from y to vi for 1?i?m. The m-competition index is a generalization of the scrambling index and the exponent of a primitive digraph. In this study, we determine an upper bound on the m-competition index of a primitive digraph using Boolean rank and give examples of primitive Boolean matrices that attain the bound.  相似文献   

16.
In [B.M. Kim, B.C. Song, W. Hwang, Primitive graphs with given exponents and minimum number of edges, Linear Algebra Appl. 420 (2007) 648-662], the minimum number of edges of a simple graph on n vertices with exponent k was determined. In this paper, we completely determine the minimum number, H(n,k), of arcs of primitive non-powerful symmetric loop-free signed digraphs on n vertices with base k, characterize the underlying digraphs which have H(n,k) arcs when k is 2, nearly characterize the case when k is 3 and propose an open problem.  相似文献   

17.
    
If G is a connected undirected simple graph on n vertices and n+c-1 edges, then G is called a c-cyclic graph. Specially, G is called a tricyclic graph if c=3. Let Δ(G) be the maximum degree of G. In this paper, we determine the structural characterizations of the c-cyclic graphs, which have the maximum spectral radii (resp. signless Laplacian spectral radii) in the class of c-cyclic graphs on n vertices with fixed maximum degree . Moreover, we prove that the spectral radius of a tricyclic graph G strictly increases with its maximum degree when , and identify the first six largest spectral radii and the corresponding graphs in the class of tricyclic graphs on n vertices.  相似文献   

18.
Wilf’s eigenvalue upper bound on the chromatic number is extended to the setting of digraphs. The proof uses a generalization of Brooks’ Theorem to digraph colorings.  相似文献   

19.
In this paper, we introduce the absolute value equations associated with second order cones (SOCAVE in short), which is a generalization of the absolute value equations discussed recently in the literature. It is proved that the SOCAVE is equivalent to a class of second order cone linear complementarity problems (SOCLCP in short). In particular, we propose a generalized Newton method for solving the SOCAVE and show that the proposed method is globally linearly and locally quadratically convergent under suitable assumptions. We also report some preliminary numerical results of the proposed method for solving the SOCAVE and the SOCLCP, which show the efficiency of the proposed method.  相似文献   

20.
In this paper, we investigate how the algebraic connectivity of a connected graph behaves when the graph is perturbed by separating or grafting an edge.  相似文献   

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