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Linear algebra. (English) Zbl 0878.05059
Beineke, Lowell W. (ed.) et al., Graph connections. Relationships between graph theory and other areas of mathematics. Based on a conference in Milton Keynes, IN, USA, 1994. Oxford: Oxford University Press. Oxf. Lect. Ser. Math. Appl. 5, 86-99 (1997).
The underlying problem considered in this chapter is the relation between the structure of a graph and various algebraic invariants, in particular, the spectrum of its adjacency matrix. The author describes necessary and sufficient conditions for a graph to be bipartite, regular, or strongly regular, for example, in terms of the spectrum. He then describes the use of linear algebra in developing feasibility tests for the existence of distance-regular graphs with given intersection arrays. The limitation of the information provided by the spectrum leads to the study of other invariants, in particular, to coordinate-free geometric attributes of eigenspaces. Certain structural properties of a graph can be expressed in terms of the eigenvalues and the angles between the coordinate axes and the eigenspaces. Finally, the author presents some results involving star partitions which give relations between individual eigenvalues and the structure of a graph.
For the entire collection see [Zbl 0862.00008].

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
15A18 Eigenvalues, singular values, and eigenvectors