An algorithm to analyse the polynomial deck of the line graph of a triangle-free graph

An algorithm is presented in which a polynomial deck, 'P D, consisting of m polynomials of degree m-1, is analysed to check whether it is the deck of characteristic polynomials of the one-vertex-deleted sub graphs of the line graph, H, of a triangle-free graph, G. We show that if two necessary conditions on 'P D, identified by counting the edges and triangles in H, are satisfied, then one can construct potential triangle-free root graphs, G, and by comparing the polynomial decks of the line graph of each with 'P D, identify the root graph.peer-reviewe

On the coefficient of λ in the characteristic polynomial of singular graphs

A singular graph, with adjacency matrix A and one zero eigenvalue, has a corresponding eigenvector v0 which is related to L, the coefficient of λ of the characteristic polynomial φ(G, λ) = Det(λI-A). In this paper a simple formula is derived expressing L in terms of the norm of v0. Furthermore it is shown that the ratio of the diagonal cofactors, which are the determinants of the adjacency matrices of the vertex-deleted subgraphs of G, can be obtained from a kernel eigenvector. The non-singular vertex-deleted subgraphs of G are characterised. Results are also obtained for singular graphs with more than one zero eigenvalue.peer-reviewe

On the Displacement of Eigenvalues when Removing a Twin Vertex

Twin vertices of a graph have the same open neighbourhood. If they are not adjacent, then they are called duplicates and contribute the eigenvalue zero to the adjacency matrix. Otherwise they are termed co-duplicates, when they contribute $-1$ as an eigenvalue of the adjacency matrix. On removing a twin vertex from a graph, the spectrum of the adjacency matrix does not only lose the eigenvalue $0$ or $-1$. The perturbation sends a rippling effect to the spectrum. The simple eigenvalues are displaced. We obtain a closed formula for the characteristic polynomial of a graph with twin vertices in terms of two polynomials associated with the perturbed graph. These are used to obtain estimates of the displacements in the spectrum caused by the perturbation

On the main eigenvalues of universal adjacency matrices and u-controllable graphs

A universal adjacency matrix U of a graph G is a linear combination of the 0–1 adjacency matrix A, the diagonal matrix of vertex degrees D, the identity matrix I and the matrix J each of whose entries is 1. A main eigenvalue of U is an eigenvalue having an eigenvector that is not orthogonal to the all–ones vector. It is shown that the number of distinct main eigenvalues of U associated with a simple graph G is at most the number of orbits of any automorphism of G. The definition of a U–controllable graph is given using control–theoretic techniques and several necessary and sufficient conditions for a graph to be U–controllable are determined. It is then demonstrated that U–controllable graphs are asymmetric and that the converse is false, showing that there exist both regular and non–regular asymmetric graphs that are not U–controllable for any universal adjacency matrix U. To aid in the discovery of these counterexamples, a gamma–Laplacian matrix L(gamma) is used, which is a simplified form of U. It is proved that any U-controllable graph is a L(gamma)–controllable graph for some parameter gamma.peer-reviewe

Minimal configurations and interlacing

A graph is singular of nullity n if zero is an eigenvalue of its adjacency matrix with multiplicity n. A subgraph that forces a graph to be singular is called a minimal configuration. We show various properties of minimal configurations.peer-reviewe