BIRS Workshop 11w5033 Linear Algebraic Techniques in Combinatorics & Graph Theory

Linear Algebra and Matrix Theory provide one of the most important tools–sometimes the only tool–in Combinatorics and Graph Theory. Even though the ideas used in applications of linear algebra to combinatorics may be very simple, the results obtained can be very strong and surprising. A famous instance is the GrahamPollak theorem which asserts that if the complete graph of order n is partitioned into m complete bipartite subgraphs, then m is at least n-1 (n-1 arises naturally by recursively deleting a star at a vertex, but there are many other ways to achieve n-1). The only known proofs of this theorem use some form of linear algebra. How does linear algebra enter into this and other combinatorial problems? Almost all combinatorial objects can be described by incidence matrices (e.g. combinatorial designs) or adjacency matrices (e.g. graphs and digraphs). Sometimes the Laplacian and Seidel matrices are also used. Therefore one basic approach is to investigate combinatorial objects by using linear algebraic parameters (rank, determinant, spectrum, etc.) of their corresponding matrices. Two of the great pioneers of this approach were H.J. Ryser and D.R. Fulkerson. Strong characterization and non-existence results can be obtained in this way. The application of linear algebra to combinatorics works in the reverse order as well. Many linear algebraic issues can be refined using combinatorial or graph-theoretic ideas. A classical instance of this is contained in the Perron-Frobenius theory of nonnegative matrices where use of an associated digraph gives more detailed information on the spectrum of the matrix. Another more recent, but now classical, instance is the use of an associated digraph to refine the classical eigenvalue inclusion region of Gershgorin. Another classical instance of the use of linear algebra to get combinatorial information is the theorem of Bruck and Ryser which rules out the existence of finite projective planes of certain orders. A further instance is the Friendship Thoerem which states that a graph in which every pair of vertices have exactly one common neighbor is a bunch of triangles, glued together in one single vertex. The important class of highly structured graphs known as strongly regular (connected) graphs have a linear algebraic characterization: they are the graphs whose adjacency matrix have exactly three distinct eigenvalues. The Graham-Pollak theorem