The theory of graph spectra can, in a way, be considered as an attempt to utilize linear algebra including, in particular, the well-developed theory of matrices for the purposes of graph theory and its applications. However, that does not mean that the theory of graph spectra can be reduced to the theory of matrices; on the contrary, it has its own characteristic features and specific ways of reasoning that fully justifying it to be treated as a theory in its own right.

It has the curious feature that some of the main results, although purely combinatorial in character, seem in the present state of knowledge to be unobtainable without resorting to algebraic methods involving a consideration of eigenvalues of adjacency matrices of graphs. There are unexplored and semi-explored territories in graph theory. It will be apparent that the results achieved so far barely scratch the surface of what appears to be a rich area of investigation.

		CARIOCA GRAPH a split nonthreshold Laplacian integral graph.

ORIGINS OF THE THEORY OF GRAPH SPECTRA

The first paper was in quantum chemistry:
* E. Hückel, Quantentheoretische Beitrage zum Benzolproblem, Z. Phys. 70(1931), 204-286

The first mathematical paper:
* L. Collatz, U. Sinogowitz, Spektren endlicher Grafen, Abh. Math. Sem. Univ. Hamburg, 21(1957), 63--77

Many years later was recognized that the two theories are essentially the same.