Francis K. Bell

A note on the irregularity of graphs

Abstract Denote by λ 1 ( G ) the largest eigenvalue of a real (0, 1)-adjacency matrix of a graph G , and by d ( G ) the mean degree of G . Collatz and Sinogowitz proposed λ 1 ( G )– d ( G ) as a measure of irregularity of G . A second such measure is the variance of the vertex degrees of G . The most irregular graphs according to these measures are determined for certain classes of graphs, and the two measures are shown to be incompatible for some pairs of graphs.

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.

On the maximal index of connected graphs

Abstract Let H (n, e) denote the set of all connected graphs having n vertices and e edges. The graphs in H (n, n + k) with maximal index are determined for k of form ( r 2 )−1 and n arbitrary.

Characterizing line graphs by star complements

Abstract In this paper it is shown that, for any odd integer t >3, the line graph L ( K t ) is the unique maximal graph having the cycle C t as a star complement for the eigenvalue −2. This result yields a characterization of L ( G ) for Hamiltonian graphs G with an odd number of vertices. We also show that, if t = r + s , where r and s are odd integers >1, then, provided that t ≠8, L ( K t ) is the unique maximal graph having C r ∪ C s as a star complement for the eigenvalue −2.

On the index of tricyclic Hamiltonian graphs

Among the tricyclic Hamiltonian graphs with a prescribed number of vertices, the unique graph with maximal index is determined. Some subsidiary results are also included.

Some additions to the theory of star partitions of graphs

This paper contains a number of results in the theory of star partitions of graphs. We illustrate a variety of situations which can arise when the Reconstruction Theorem for graphs is used, considering in particular galaxy graphs — these are graphs in which every star set is independent. We discuss a recursive ordering of graphs based on the Reconstruction Theorem, and point out the significance of galaxy graphs in this connection.

Graph eigenspaces of small codimension

Abstract For a given positive integer t there are only finitely many graphs with an eigenvalue μ ∉{−1,0} such that the eigenspace of μ has codimension t. The graphs for which t ⩽5 are identified.

A Note on the Second Largest Eigenvalue of Star-Like Trees

Star-like trees axe trees homeomorphic to stars. In this paper we identify those star-like trees for which the second largest eigenvalue is extremal — either minimal or maximal — when certain conditions are imposed. We also obtain partial results on the way in which the second largest eigenvalue of a simple class of star-like trees changes under local modifications (graph perturbations). Analogous problems for the largest eigenvalue (known as the index of the graph) have been widely studied in the literature.