6097933
The Clar covering polynomial of hexagonal systems III
Abstract The Clar covering polynomial is a recently proposed concept of hexagonal systems by which some important topological indices such as perfect matching count, Clar number, first Herndon number, etc., can be easily obtained. In this paper we establish a relationship between the Clar covering polynomial and sextet polynomial. A lower bound of the Clar number and some properties of coefficients of the Clar covering polynomial are thus deduced. It is mentioned that the summation of coefficients of Clar covering polynomial can be used to calculate the number of perfect matchings of a certain kind of polyominoes relating to crystal physics.
34749580
Plane elementary bipartite graphs
A connected graph is called elementary if the union of all perfect matchings forms a connected subgraph. In this paper we mainly study various properties of plane elementary bipartite graphs so that many important results previously obtained for hexagonal systems are treated in a unified way. Firstly, we show that a plane bipartite graph G is elementary if and only if the boundary of each face (including the infinite face) is an alternating cycle with respect to some perfect matching of G. For a plane bipartite graph G all interior vertices of which are of the same degree, a stronger result is obtained; namely, G is elementary if and only if the boundary of the infinite face of G is an alternating cycle with respect to some perfect matching of G. Second, the concept of the Z-transformation graph Z(G) of a hexagonal system G (whose vertices represent the perfect matchings of G) is extended to a plane bipartite graph G and some results analogous to those for hexagonal systems are obtained. A peripheral face f of G is called reducible if the removal of the internal vertices and edges of the path that is the intersection of f and the exterior face of G results in a plane elementary bipartite graph. Thirdly, we obtain the reducible face decomposition for plane elementary bipartite graphs. Furthermore, sharp upper and lower bounds for the number of reducible faces are derived. Conversely, we can construct any plane elementary bipartite graphs by adding new peripheral faces one by one. As applications of this approach, we give simple construction methods for several types of plane elementary bipartite graphs G that contain a forcing edge (which belongs to exactly one perfect matching of G) and whose Z-transformation graphs Z(G) contain vertices of degree one.
204989114
On the numbers of spanning trees and Eulerian tours in generalized de Bruijn graphs
Abstract In this paper, we give the spectra and characteristic polynomials of generalized de Bruijn graphs. By Tutte's Theorem we can obtain the number of spanning trees and the number of Eulerian tours in such graphs.
21190705
Evolving Spanning Trees Using the Heat Equation
This paper explores how to use the heat kernel to evolve the minimum spanning tree of a graph with time. We use the heat kernel to weight the edges of the graph, and these weights can be computed by exponentiating the Laplacian eigensystem of the graph with time. The resulting spanning trees exhibit an interesting behaviour as time increases. Initially, they are bushy and rooted near the centre of graph, but as time evolves they become string-like and hug the boundary of the graph. We characterise this behaviour using the distribution of terminal nodes with time, and use this distribution for the purposes of graph clustering and image segmentation.
14205748
Edge Detection and Anisotropic Diffusion for Tensor-Valued Images
This paper considers feature space of DT-MRI as a differential manifold with an affine-invariant metric, where we generalise Di Zenzo's structure tensor to tensor-valued images for edge detection. To improve the quality of the edges, we smooth out the noise by generalised Perona-Malik method for tensor images. We demonstrate our algorithm on both synthetic and real DT-MRI data.
40411122
Super Restricted Edge Connectivity of Regular Graphs
An edge cut of a connected graph is called restricted if it separates this graph into components each having order at least 2; a graph G is super restricted edge connected if G−S contains an isolated edge for every minimum restricted edge cut S of G. It is proved in this paper that k-regular connected graph G is super restricted edge connected if k > |V(G)|/2+1. The lower bound on k is exemplified to be sharp to some extent. With this observation, we determined the number of edge cuts of size at most 2k−2 of these graphs.
32475390
k-Cycle resonant graphs
Abstract A connected graph G is said to be k -cycle resonant if, for 1 ⩽ t ⩽ k , any t disjoint cycles in G are mutually resonant, that is, there is a perfect matching M of G such that each of the t cycles is an M -alternating cycle. In this paper, we at the first time introduce the concept of k -cycle resonant graphs, and investigate some properties of k -cycle resonant graphs. Some simple necessary and sufficient conditions for a graph to be k -cycle resonant are given. The construction of k -cycle resonant hexagonal systems are also characterized.