Fuji Zhang

Resistance distance and the normalized Laplacian spectrum

It is well known that the resistance distance between two arbitrary vertices in an electrical network can be obtained in terms of the eigenvalues and eigenvectors of the combinatorial Laplacian matrix associated with the network. By studying this matrix, people have proved many properties of resistance distances. But in recent years, the other kind of matrix, named the normalized Laplacian, which is consistent with the matrix in spectral geometry and random walks [Chung, F.R.K., Spectral Graph Theory, American Mathematical Society: Providence, RI, 1997], has engendered peoples attention. For many people think the quantities based on this matrix may more faithfully reflect the structure and properties of a graph. In this paper, we not only show the resistance distance can be naturally expressed in terms of the normalized Laplacian eigenvalues and eigenvectors of G, but also introduce a new index which is closely related to the spectrum of the normalized Laplacian. Finally we find a non-trivial relation between the well-known Kirchhoff index and the new index.

On acyclic conjugated molecules with minimal energies

Abstract The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph. In [5] Gutman (J. Math. Chem. 1 (1987) 123–143) proposes two conjectures about the minimum of the energy of conjugated trees (trees with a perfect matching). This paper mathematically verifies the two conjectures. In addition, trees with the second and the third smallest energies are also discussed.

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.

On the ordering of graphs with respect to their matching numbers

Abstract The number of k-matchings in a graph G is denoted by m(G,k). If for two graphs G1 and G2, m(G1, k) ≥m(G2,k) for all k, then we write G1 > G2. Six classes of graphs are ordered with respect to the relation >.

The Clar covering polynomial of hexagonal systems I

Abstract In this paper the Clar covering polynomial of a hexagonal system is introduced. In fact it is a kind of F polynomial [4] of a graph, and can be calculated by recurrence relations. We show that the number of aromatic sextets (in a Clar formula), the number of Clar formulas, the number of Kekule structures and the first Herndon number for any Kekulean hexagonal system can be easily obtained by its Clar covering polynomial. In addition, we give some theorems to calculate the Clar covering polynomial of a hexagonal system. As examples we finally derive the explicit expressions of the Clar covering polynomials for some small hexagonal systems and several types of catacondensed hexagonal systems. A relation between the resonance energy and the Clar covering polynomial of a hexagonal system is considered in the next paper.

When each hexagon of a hexagonal system covers it

Abstract In this paper we establish a simple criterion which enables us to determine whether or not a hexagonal system H has the property that each hexagon s of H is resonant, i.e., the subgraph obtained by deleting from H the vertices of s together with their edges has at least one perfect matching.

Hexagonal systems with forcing edges

An edge of a hexagonal system H is said to be forcing if it belongs to exactly one perfect matching of H. Using the concept of Z-transformation of hexagonal system, we give a characterization for the hexagonal systems with forcing edges and determine all forcing edges is such systems. We also give the generating function of all hexagonal systems with forcing edges. A hexaoonal system, also called hexanimal, polyhex or benzenoid system [2-4], is a finite connected planar graph without cut vertices in which every interior region is surrounded by a regular hexagon of side length 1. Recently, three books have been published on this kind of systems. The concept of forcing edges was first defined in [7], which related to some chemical and physical problems, see [8-10], namely, the innate degree of freedom of ;t-electron couplings and long-range order for spin pairing. We first give the following definition.

New lower bound on the number of perfect matchings in fullerene graphs

A fullerene graph is a cubic and 3-connected plane graph (or spherical map) that has exactly 12 faces of size 5 and other faces of size 6, which can be regarded as the molecular graph of a fullerene. T. Došlić [3] obtained that a fullerene graph with p vertices has at least (p+2)/2 perfect matchings by applying the recently developed decomposition techniques in matching theory of graphs. This note gets a better lower bound ⌈3(p+2)/4⌉ of the number of perfect matchings of a fullerene graph by finding its 2-extendability. This property further implies a chemical consequence that every derivative of a fullerene by substituting any two pairs of adjacent carbon atoms permits a Kekulé structure.

Hexagonal chains with minimal total π-electron energy

Abstract The total π -electron energy of a conjugated molecule (within the framework of HMO approximation) can be calculated by its molecular graph. For the acyclic molecules, the extremal problem of some important types of conjugated molecules have been solved previously. This Letter for the first time deals with a type of cyclic conjugated molecule–benzenoid chains (hexagonal chains). We prove that in the set of all hexagonal chains with n hexagons, the linear polyacene has the minimal energy. Furthermore, a sharp lower bound of total π -electron energies of the hexagonal chain is also obtained.

Hexagonal chains with maximal total π-electron energy

Abstract In the preceding Letter we determined the hexagonal chain (benzenoid hydrocarbon) with minimal total π-electron energy. Further to that work, in this Letter the following result is obtained: in the set of all hexagonal chains with n hexagons, the zig-zag chain (zig-zag polyacene) has the maximal energy.