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991.
Let G be a graph on n vertices, and let λ1,λ2,…,λn be its eigenvalues. The Estrada index is defined as . We determine the unique tree with maximum Estrada index among the trees on n vertices with given matching number, and the unique tree with maximum Estrada index among the trees on n vertices with fixed diameter. For , we also determine the tree with maximum Estrada index among the trees on n vertices with maximum degree Δ. It gives a partial solution to the conjecture proposed by Ili? and Stevanovi? in Ref. [14]. 相似文献
992.
David Dol?an 《Linear algebra and its applications》2011,435(7):1657-1665
We calculate diameters and girths of commuting graphs of the set of all nilpotent matrices over a semiring, the group of all invertible matrices over a semiring, and the full matrix semiring. 相似文献
993.
In this paper, we shall give a survey of applications of the theory of graph spectra to Computer Science. Eigenvalues and eigenvectors of several graph matrices appear in numerous papers on various subjects relevant to information and communication technologies. In particular, we survey applications in modeling and searching Internet, in computer vision, data mining, multiprocessor systems, statistical databases, and in several other areas. Some related new mathematical results are included together with several comments on perspectives for future research. In particular, we claim that balanced subdivisions of cubic graphs are good models for virus resistent computer networks and point out some advantages in using integral graphs as multiprocessor interconnection networks. 相似文献
994.
Let Cn,g be the lollipop graph obtained by appending a g-cycle Cg to a pendant vertex of a path on n-g vertices. In 2002, Fallat, Kirkland and Pati proved that for and g?4, α(Cn,g)>α(Cn,g-1). In this paper, we prove that for g?4, α(Cn,g)>α(Cn,g-1) for all n, where α(Cn,g) is the algebraic connectivity of Cn,g. 相似文献
995.
Huiqiu Lin 《Linear algebra and its applications》2011,434(12):2462-2467
Let D be a digraph with vertex set V(D). A partition of V(D) into k acyclic sets is called a k-coloring of D. The minimum integer k for which there exists a k-coloring of D is the dichromatic number χ(D) of the digraph D. Denote Gn,k the set of the digraphs of order n with the dichromatic number k≥2. In this note, we characterize the digraph which has the maximal spectral radius in Gn,k. Our result generalizes the result of [8] by Feng et al. 相似文献
996.
We define a family of weighted geometric means {G(t;ω;A)}t∈[0,1]n where ω and A vary over all positive probability vectors in Rn and n-tuples of positive definite matrices resp. Each of these weighted geometric means interpolates between the weighted ALM (t=0n) and BMP (t=1n) geometric means (ALM and BMP geometric means have been defined by Ando-Li-Mathias and Bini-Meini-Poloni, respectively.) We show that the weighted geometric means satisfy multidimensional versions of all properties that one would expect for a two-variable weighted geometric mean. 相似文献
997.
Given an n-vertex graph G=(V,E), the Laplacian spectrum of G is the set of eigenvalues of the Laplacian matrix L=D-A, where D and A denote the diagonal matrix of vertex-degrees and the adjacency matrix of G, respectively. In this paper, we study the Laplacian spectrum of trees. More precisely, we find a new upper bound on the sum of the k largest Laplacian eigenvalues of every n-vertex tree, where k∈{1,…,n}. This result is used to establish that the n-vertex star has the highest Laplacian energy over all n-vertex trees, which answers affirmatively to a question raised by Radenkovi? and Gutman [10]. 相似文献
998.
Zejun Huang 《Linear algebra and its applications》2011,434(4):880-883
A sign pattern is said to be nilpotent of index k if all real matrices in its qualitative class are nilpotent and their maximum nilpotent index equals k. In this paper, we characterize sign patterns that are nilpotent of a given index k. The maximum number of nonzero entries in such sign patterns of a given order is determined as well as the sign patterns with this maximum number of nonzero entries. 相似文献
999.
For a given graph G with (0, 1)-adjacency matrix AG, the generalized characteristic polynomial of G is defined to be ?G=?G(λ,t)=det(λI-(AG-tDG)), where I is the identity matrix and DG is the diagonal degree matrix of G. In this paper, we are mainly concerned with the problem of characterizing a given graph G by its generalized characteristic polynomial ?G. We show that graphs with the same generalized characteristic polynomials have the same degree sequence, based on which, a unified approach is proposed to show that some families of graphs are characterized by ?G. We also provide a method for constructing graphs with the same generalized characteristic polynomial, by using GM-switching. 相似文献
1000.
Bao-Xuan Zhu 《Linear algebra and its applications》2011,434(9):2030-2041
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. 相似文献