Bojan Mohar

Some applications of Laplace eigenvalues of graphs

In the last decade important relations between Laplace eigenvalues and eigenvectors of graphs and several other graph parameters were discovered. In these notes we present some of these results and discuss their consequences. Attention is given to the partition and the isoperimetric properties of graphs, the max-cut problem and its relation to semidefinite programming, rapid mixing of Markov chains, and to extensions of the results to infinite graphs.

Isoperimetric numbers of graphs

Abstract For X ⊆ V ( G ), let ∂X denote the set of edges of the graph G having one end in X and the other end in V ( G ) βX . The quantity i(G)≔ min { |∂X| |X| } , where the minimum is taken over all non-empty subsets X of V ( G ) with |X| ≤ |V(G)| 2 , is called the isoperimetric number of G . The basic properties of i ( G ) are discussed. Some upper and lower bounds on i ( G ) are derived, one in terms of | V ( G )| and | E ( G )| and two depending on the second smallest eigenvalue of the difference Laplacian matrix of G . The upper bound is a strong discrete version of the wellknown Cheeger inequality bounding the first eigenvalue of a Riemannian manifold. The growth and the diameter of a graph G are related to i ( G ). The isoperimetric number of Cartesian products of graphs is studied. Finally, regular graphs of fixed degree with large isoperimetric number are considered.

Eigenvalues, diameter, and mean distance in graphs

It is well-known that the second smallest eigenvalueλ2 of the difference Laplacian matrix of a graphG is related to the expansion properties ofG. A more detailed analysis of this relation is given. Upper and lower bounds on the diameter and the mean distance inG in terms ofλ2 are derived.

Laplace eigenvalues of graphs—a survey

Abstract Several applications of Laplace eigenvalues of graphs in graph theory and combinatorial optimization are outlined.

The Quasi-Wiener and the Kirchhoff Indices Coincide

In 1993 two novel distance-based topological indices were put forward. In the case of acyclic molecular graphs both are equal to the Wiener index, but both differ from it if the graphs contain cycles. One index is defined (Mohar, B.; Babic, D.; Trinajstic, N. J. Chem. Inf. Comput. Sci. 1993, 33, 153−154) in terms of eigenvalues of the Laplacian matrix, whereas the other is conceived (Klein, D. J.; Randic, M. J. Math. Chem. 1993, 12, 81−95) as the sum of resistances between all pairs of vertices, assuming that the molecule corresponds to an electrical network, in which the resistance between adjacent vertices is unity. Eventually, the former quantity was named quasi-Wiener index and the latter Kirchhoff index. We now demonstrate that the quasi-Wiener and Kirchhoff indices of all graphs coincide.

Optimal linear labelings and eigenvalues of graphs

Abstract For several NP-hard optimal linear labeling problems, including the bandwidth, the cutwidth, and the min-sum problem for graphs, a heuristic algorithm is proposed which finds approximative solutions to these problems in polynomial time. The algorithm uses eigenvectors corresponding to the second smallest Laplace eigenvalue of a graph. Although bad in some “degenerate” cases, the algorithm shows fairly good behaviour. Several upper and lower bounds on the bandwidth, cutwidth, and min-p-sums are derived. Most of these bounds are given in terms of Laplace eigenvalues of the graphs. They are used in the analysis of our algorithm and as measures for the error of the obtained approximation to an optimal labeling.

A Linear Time Algorithm for Embedding Graphs in an Arbitrary Surface

For an arbitrary fixed surface S, a linear time algorithm is presented that for a given graph G either finds an embedding of G in S or identifies a subgraph of G that is homeomorphic to a minimal forbidden subgraph for embeddability in S. A side result of the proof of the algorithm is that minimal forbidden subgraphs for embeddability in S cannot be arbitrarily large. This yields a constructive proof of the result of Robertson and Seymour that for each closed surface there are only finitely many minimal forbidden subgraphs. The results and methods of this paper can be used to solve more general embedding extension problems.

Eigenvalues in Combinatorial Optimization

In the last decade many important applications of eigenvalues and eigenvectors of graphs in combinatorial optimization were discovered. The number and importance of these results is so fascinating that it makes sense to present this survey.

How to compute the Wiener index of a graph

The Wiener index of a graphG is equal to the sum of distances between all pairs of vertices ofG. It is known that the Wiener index of a molecular graph correlates with certain physical and chemical properties of a molecule. In the mathematical literature, many good algorithms can be found to compute the distances in a graph, and these can easily be adapted for the calculation of the Wiener index. An algorithm that calculates the Wiener index of a tree in linear time is given. It improves an algorithm of Canfield, Robinson and Rouvray. The question remains: is there an algorithm for general graphs that would calculate the Wiener index without calculating the distance matrix? Another algorithm that calculates this index for an arbitrary graph is given.

Labeling of Benzenoid Systems which Reflects the Vertex-Distance Relations

It is shown that the vertices of benzenoid systems admit a labeling which reflects their distance relations. To every vertex of a molecular graph of a benzenoid hydrocarbon a sequence of zeros and ones (a binary number) can be associated, such that the number of positions in which these sequences differ is equal to the graph-theoretic vertex distance. It is shown by an example that such labelings can be used not only for nomenclature purposes but also for fast evaluation of molecular parameters based on the graph distance.