Peter Rowlinson

The largest eigenvalue of a graph: A survey

This article is a survey of results concerning the largest eigenvalue (or index) of a grapn, catcgoiizeu as follows (1) inequalities lor the index, (2) graph with bounded index, (3) ordering graphs by their indices, (4) graph operations and modifications, (5) random graphs, (6) applications.

Spectra of unicyclic graphs

Unicyclic graphs are discussed in the context of graph orderings related to eigenvalues. Several theorems involving lexicographical ordering by spectral moments as well as the ordering by the largest eigenvalue are proved. An appendix contains a table of the 89 unicyclic graphs on eight vertices together with their spectra, spectral moments and characteristic polynomials.

The main eigenvalues of a graph: a survey

We survey results relating main eigenvalues and main angles to the structure of a graph. We provide a number of short proofs, and note the connection with star partitions. We discuss graphs with just two main eigenvalues in the context of measures of irregularity, and in the context of harmonic graphs.

On the maximal index of graphs with a prescribed number of edges

Abstract Among the graphs with a prescribed number of edges, those with maximal index are determined. The result confirms a conjecture of Brualdi and Hoffman.

Variable neighborhood search for extremal graphs. 16. Some conjectures related to the largest eigenvalue of a graph

We consider four conjectures related to the largest eigenvalue of (the adjacency matrix of) a graph (i.e., to the index of the graph). Three of them have been formulated after some experiments with the programming system AutoGraphiX, designed for finding extremal graphs with respect to given properties by the use of variable neighborhood search. The conjectures are related to the maximal value of the irregularity and spectral spread in n-vertex graphs, to a Nordhaus-Gaddum type upper bound for the index, and to the maximal value of the index for graphs with given numbers of vertices and edges. None of the conjectures has been resolved so far. We present partial results and provide some indications that the conjectures are very hard.

On the Multiplicities of Graph Eigenvalues

Star complements and associated quadratic functions are used to obtain a sharp upper bound for the order of a graph with an eigenspace of prescribed codimension. It is shown that for regular graphs the bound can be reduced by 1, and that this reduced bound is attained by a regular graph G if and only if G is an extremal strongly regular graph. 2000 Mathematics Subject Classification 05C50.

Graphs with Least Eigenvalue −2: The Star Complement Technique

Let G be a connected graph with least eigenvalue −2, of multiplicity k. A star complement for −2 in G is an induced subgraph H = G − X such that |X| = k and −2 is not an eigenvalue of H. In the case that G is a generalized line graph, a characterization of such subgraphs is used to decribe the eigenspace of −2. In some instances, G itself can be characterized by a star complement. If G is not a generalized line graph, G is an exceptional graph, and in this case it is shown how a star complement can be used to construct G without recourse to root systems.

A study of eigenspaces of graphs

Abstract We investigate the relationship between the structure of a graph and its eigenspaces. The angles between the eigenspaces and the vectors of the standard basis of R n play an important role. The key notion is that of a special basis for an eigenspace called a star basis. Star bases enable us to define a canonical bases of R n associated with a graph, and to formulate an algorithm for graph isomorphism.

Constructing fullerene graphs from their eigenvalues and angles

We discuss means of constructing fullerene graphs from their eigenvalues and angles. An algorithm for such a construction is given.

The maximal exceptional graphs

A graph is said to be exceptional if it is connected, has least eigenvalue greater than or equal to -2, and is not a generalized line graph. Such graphs are known to be representable in the exceptional root system E8. We determine the maximal exceptional graphs by a computer search using the star complement technique, and then show how they can be found by theoretical considerations using a representation of E8 in R8. There are exactly 473 maximal exceptional graphs.