Preface 1. Introduction 2. Graph operations and modifications 3. Spectrum and structure 4. Characterizations by spectra 5. Structure and one eigenvalue 6. Spectral techniques 7. Laplacians 8. Additional topics 9. Applications Appendix Bibliography Index of symbols Index.

An Introduction to the Theory of Graph Spectra: Spectral techniques

A. Sinclair and M. Jerrum (1988) derived a bound on the mixing rate of time-reversible Markov chains in terms of their conductance. The authors generalize this result by not assuming time reversibility and using a weaker notion of conductance. They prove an isoperimetric inequality for subsets of a convex body. These results are combined to simplify an algorithm of M. Dyer et al. (1989) for approximating the volume of a convex body and to improve running-time bounds. >

The mixing rate of Markov chains, an isoperimetric inequality, and computing the volume

On the topological index

Minimizing Kirchhoff index among graphs with a given vertex bipartiteness

The resistance distance rij between vertices i and j of a connected (molecular) graph G is computed as the effective resistance between nodes i and j in the corresponding network constructed from G by replacing each edge of G with a unit resistor. The Kirchhoff index Kf(G) is the sum of resistance distances between all pairs of vertices. In this work, according to the decomposition theorem of Laplacian polynomial, we obtain that the Laplacian spectrum of linear hexagonal chain Ln consists of the Laplacian spectrum of path P2n+1 and eigenvalues of a symmetric tridiagonal matrix of order 2n + 1. By applying the relationship between roots and coefficients of the characteristic polynomial of the above matrix, explicit closed-form formula for Kirchhoff index of Ln is derived in terms of Laplacian spectrum. To our surprise, the Krichhoff index of Ln is approximately to one half of its Wiener index. Finally, we show that holds for all graphs G in a class of graphs including Ln. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008

Kirchhoff index of linear hexagonal chains

Resistance distance and the normalized Laplacian spectrum

Random walks and the effective resistance sum rules

On extremal bipartite unicyclic graphs

In [D.J. Klein, Croat. Chem. Acta. 75(2), 633 (2002)] Klein established a number of sum rules to compute the resistance distance of an arbitrary graph, especially he gave a specific set of local sum rules that determined all resistance distances of a graph (saying the set of local sum rules is complete). Inspired by this result, we give another complete set of local rules, which is simple and also efficient, especially for distance-regular graphs. Finally some applications to chemical graphs (for example the Platonic solids as well as their vertex truncations, which include the graph of Buckminsterfullerene and the graph of boron nitride hetero-fullerenoid B 12 N 12) are made to illustrate our approach.

Haiyan Chen

Papers

Resistance distance and the normalized Laplacian spectrum

Random walks and the effective resistance sum rules

On extremal bipartite unicyclic graphs

Resistance distance local rules

The expected hitting times for finite Markov chains