Weigen Yan

The behavior of Wiener indices and polynomials of graphs under five graph decorations

Abstract The sum of distances between all vertex pairs in a connected graph is known as the Wiener index. It is an early index which correlates well with many physico-chemical properties of organic compounds and as such has been well studied over the last quarter of a century. A q -analogue of this index, termed the Wiener polynomial by Hosoya but also known today as the Hosoya polynomial , extends this concept by trying to capture the complete distribution of distances in the graph. Mathematicians have studied several operators on a connected graph in which we see a subdivision of the edges. In this work, we show how the Wiener index of a graph changes with these operations, and extend the results to Wiener polynomials.

The matching energy of graphs with given parameters

Gutman and Wagner I. Gutman, S. Wagner, The matching energy of a graph, Discrete Appl. Math. 160 (2012) 2177-2187] defined the matching energy of a graph and gave some properties and asymptotic results of the matching energy. In this paper, we characterize the connected graph G with the connectivity ? (resp. chromatic number ? ) which has the maximum matching energy.

Asymptotic energy of lattices

The energy of a simple graph G arising in chemical physics, denoted by E(G), is defined as the sum of the absolute values of eigenvalues of G. As the dimer problem and spanning trees problem in statistical physics, in this paper we propose the energy per vertex problem for lattice systems. In general for a type of lattice in statistical physics, to compute the entropy constant with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions are different tasks with different hardness and may have different solutions. We show that the energy per vertex of plane lattices is independent of the toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions. In particular, the asymptotic formulae of energies of the triangular, 33.42, and hexagonal lattices with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions are obtained explicitly.

Enumeration of subtrees of trees

Let T be a weighted tree. The weight of a subtree T1 of T is defined as the product of weights of vertices and edges of T1. We obtain a linear-time algorithm to count the sum of weights of subtrees of T . As applications, we characterize the tree with the diameter at least d, which has the maximum number of subtrees, and we characterize the tree with the maximum degree at least ∆, which has the minimum number of subtrees.

Enumeration of perfect matchings of a type of Cartesian products of graphs

Let G be a graph and let Pm(G) denote the number of perfect matchings of G.We denote the path with m vertices by Pm and the Cartesian product of graphs G and H by G × H. In this paper, as the continuance of our paper [W. Yan, F. Zhang, Enumeration of perfect matchings of graphs with reflective symmetry by Pfaffians, Adv. Appl. Math. 32 (2004) 175-188], we enumerate perfect matchings in a type of Cartesian products of graphs by the Pfaffian method, which was discovered by Kasteleyn. Here are some of our results: 1. Let T be a tree and let Cn denote the cycle with n vertices. Then Pm(C4 × T) = Π(2 + α2), where the product ranges over all eigenvalues α of T. Moreover, we prove that Pm(C4 × T) is always a square or double a square. 2. Let T be a tree. Then Pm(P4 × T) = Π(1 + 3α2 + α2), where the product ranges over all non-negative eigenvalues α of T. 3. Let T be a tree with a perfect matching. Then Pm(P3 × T) = Π(2 + α2), where the product ranges over all positive eigenvalues α of T. Moreover, we prove that Pm(C4 × T) = [Pm(P3 × T)]2.

Graphical condensation for enumerating perfect matchings

The method of graphical condensation for enumerating perfect matchings was found by Propp (Theoret. Comput. Sci. 303 (2003) 267), and was generalized by Kuo (Theoret. Comput. Sci. 319 (2004) 29). In this paper, we obtain some more general results on graphical condensation than Kuos. Our method is also different from Kuos. As applications of our results, we obtain a new proof of Stanleys multivariate version of the Aztec diamond theorem and we enumerate perfect matchings of a type of molecular graph. Finally, a combinatorial identity on the number of plane partitions is also given.

The asymptotic behavior of some indices of iterated line graphs of regular graphs

In this paper, we consider the asymptotic behavior of the number of spanning trees and the Kirchhoff index of iterated line graphs and iterated para-line graphs (or clique-inserted graphs) of a regular graph G. We show that the asymptotic behavior of these indices (except the Kirchhoff index of the iterated para-line graphs) is independent of the structure of the regular graph G.

The number of spanning trees of plane graphs with reflective symmetry

A plane graph is called symmetric if it is invariant under the reflection across some straight line (called symmetry axis). Let G be a symmetric plane graph. We prove that if there is no edge in G intersected by its symmetry axis then the number of spanning trees of G can be expressed in terms of the product the number of spanning trees of two smaller graphs. each of which has about half the number of vertices of G.

Enumerating spanning trees of graphs with an involution

As the extension of the previous work by Ciucu and the present authors [M. Ciucu, W.G. Yan, F.J. Zhang, The number of spanning trees of plane graphs with reflective symmetry, J. Combin. Theory Ser. A 112 (2005) 105-116], this paper considers the problem of enumeration of spanning trees of weighted graphs with an involution which allows fixed points. We show that if G is a weighted graph with an involution, then the sum of weights of spanning trees of G can be expressed in terms of the product of the sums of weights of spanning trees of two weighted graphs with a smaller size determined by the involution of G. As applications, we enumerate spanning trees of the almost-complete bipartite graph, the almost-complete graph, the Mobius ladder, and the almost-join of two copies of a graph.

Graphical condensation of plane graphs: a combinatorial approach

The method of graphical vertex-condensation for enumerating perfect matchings of plane bipartite graph was found by Propp [Generalized Domino-shuffling, Theoret. Comput. Sci. 303 (2003) 267-301], and was generalized by Kuo [Applications of graphical condensation for enumerating matchings and tilings, Theoret. Comput. Sci. 319 (2004) 29-57] and Yah and Zhang [Graphical condensation for enumerating perfect matchings, J. Combin. Theory Ser. A 110 (2005) 113-125]. In this paper, by a purely combinatorial method some explicit identities on graphical vertex-condensation for enumerating perfect matchings of plane graphs (which do not need to be bipartite) are obtained. As applications of our results, some results on graphical edge-condensation for enumerating perfect matchings are proved, and we count the sum of weights of perfect matchings of weighted Aztec diamond.