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

In this paper, we define a graph operation, namely, M-edge rooted product of graphs. This generalizes the existing graph operation called graphs with edge pockets. Also we introduce a matrix invariant, namely, coronal of a matrix constrained by the index sets. We compute this value for some class of matrices with respect to some index sets. We obtain the generalized characteristic polynomial of the graph obtained by M-edge rooted product with a help of this invariant. Consequently, we deduce the characteristic polynomial of the adjacency matrix, the Laplacian matrix and the signless Laplacian matrix of this graph. Using these results, we derive the L-spectrum of several families of M-edge rooted product of graphs and deduce several existing results on the spectra of graphs with edge pockets in the literature. As applications, we obtain infinitely many L-cospectral graphs and construct A-integral graphs, L-integral graphs.

Spectra of M-edge rooted product of graphs

Spectra of M-bicone complement product of graphs

In this paper, first we consider some partitioned matrices having some special kind of blocks, and we obtain their eigenvalues and eigenvectors. From these, we deduce several results on the eigenvalues of partitioned matrices in the literature. Next we consider an extension of a generalized join of graphs introduced by Hedetniemi (On classes of graphs defined by special cutsets of lines. In: Many facets of graph theory. Proceedings of the conference held at Western MiGammagan University, Kalamazoo/Mi, 1968. Lecture notes in mathematics, vol 110, pp 171–189, 1969), we call it the $${\mathcal {M}}$$
 -join of $${\mathcal {H}}_k$$
 . The spectra of partitioned matrices allow to detect the adjacency, the Laplacian and the signless Laplacian spectrum of old and new type of join of graphs. Further, we have constructed several pairs of simultaneously A-cospectral, L-cospectral and Q-cospectral graphs.

Rajkumar Rajendran

Papers

Spectra of partitioned matrices and the $${\mathcal {M}}$$ M -join of graphs