Let G $$ be a finite tree. For any matching M of , let U ( ) U(M) the set vertices uncovered by . ℳ {\mathcal{M}}_G uniform random maximum size In this paper, we analyze structure U\left({\mathcal{M}}_G\right) We first show that is determinantal process. also for most process in small neighborhood vertex can well approximated based on somewhat larger same vertex. Then normalized Shannon entropy...

Quantum algorithms for graph problems are considered, both in the adjacency matrix model and in an adjacency list-like array model. We give almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, Strong Connectivity, Minimum Spanning Tree, and Single Source Shortest Paths. For example we show that the query complexity of Minimum Spanning Tree is in Θ(...

The characteristic polynomial of the adjacency matrix of a graph is noted in connection with a quantity characterizing the topological nature of structural isomers saturated hydrocarbons [S], a set of numbers that are the same for all graphs isomorphic to the graph, and others [l]. Many properties of the characteristic polynomials of the adjacency matrices of a graph and its line graph [3] have...

An Euclidean graph associated with a molecule is defined by a weighted graph with adjacency matrix M = [dij], where for ij, dij is the Euclidean distance between the nuclei i and j. In this matrix dii can be taken as zero if all the nuclei are equivalent. Otherwise, one may introduce different weights for distinct nuclei. Balaban introduced some monster graphs and then Randic computed complexit...

A mixed graph is called second kind hermitian integral (or HS-integral) if the eigenvalues of its Hermitian-adjacency matrix are integers. Eisenstein (0, 1)-adjacency Let ? be an abelian group. We characterize set S for which a Cayley Cay(?,S) HS-integral. also show that and only it

Although K-Means clustering has been widely studied due to its simplicity, these methods still have the following fatal drawbacks. Firstly, they need initialize cluster centers, which causes unstable performance. Secondly, poor performance on non-Gaussian datasets. Inspired by affinity matrix, we propose a novel multi-view based adjacency matrix. It maps matrix distance according principle that...

For a set Ω of size n ≥ 7 and Ω{3} the set of subsets of Ω of size 3, we examine the binary codes obtained from the adjacency matrix of each of the three graphs with vertex set Ω{3}, with adjacency defined by two vertices as 3-sets being adjacent if they have zero, one or two elements in common, respectively.

6.8. Protein Profiles 823 7. Spectrum-like Representation of Proteins 824 8. Graphical Approach to Protein Alignment 825 9. Nongraphical Representation of Proteins 828 9.1. Representation of Proteins as Walks in Space 828 9.2. Protein Descriptors 829 10. Most Recent Results 830 10.1. Amino Acid Adjacency-Count Matrix 830 10.2. Higher-Order AA Adjacency-Count Matrices 834 10.3. Simplified Graphi...

We give short proofs of the adjacency matrix characterizations of interval digraphs and unit interval digraphs.