The inuence of Miroslav Fiedler on spectral graph theory

This note is a write up of a talk given at the ILAS meeting in Braunschweig, 2011, at the minisymposium celebrating the 80th birthday of Miroslav Fiedler. The purpose of the talk is to outline the impact of Fiedler’s work on the development of spectral graph theory. Fiedler is best known for putting forward the algebraic connectivity and its eigenvector. These two topics were genuine gold strikes which have motivated thousands of studies in pure and applied science. But there is more in Fiedler’s work that in‡uences spectral graph theory, and still more waiting to be discovered. Keywords: algebraic connectivity, Fiedler vector, elliptic matrix, additive compound matrix. AMS classi…cation: 05C50, 05C35 About gold strikes in mathematics In mathematics, as in life, there are people that can strike gold. When real gold has been struck, we can see that, as inevitably a gold rush follows. Since the times of Euclid there have been numerous golds strikes in mathematics, but here are two more recent examples that caused memorable rushes: the semi-circle law of Wigner from 1955 about the distribution of eigenvalues of random real symmetric matrices and the regularity lemma of Szemerédi from 1978 about the structure of large graphs. Thus, when we use here “gold in mathematics”this is not about monumental theories but about short and sharp assertions that can attract people to follow overnight. Unfortunately, in mathematics gold is struck as seldom as in life, and few are the lucky ones that are able to do it. This ability is indeed uncanny, it is not the skill or even imagination that matters, one truly needs prospector’s eyes and hunch. Today we are speaking about the work of Miroslav Fiedler, a mathematician who struck gold on more than one occasion. His brilliant contributions to matrix theory, geometry, numerical methods Department of Mathematical Sciences, University of Memphis, Memphis TN 38152, USA; email: [email protected] yResearch supported by NSF Grant DMS-0906634.