Some results from algebraic graph theory - Azarija, Jernej (visokošolska dela, 2016)
In this thesis we present some results living in the intersection between graph theory and linear algebra. We introduce the subject of algebraic graph theory presenting some general results from this area. In particular we show how certain algebraic objects such as matrices and polynomials can be used to gain structural information about graphs. We then introduce two graph polynomials namely the chromatic polynomial and its generalization - the Tutte polynomial. We present a counterexample to a conjecture of J. Xu and Z. Liu about the chromatic polynomial and degree sequences. We then turn our attention to matrices associated with graphs namely the adjacency matrix and distance matrix. We present some results in the context of strongly regular graphs. In particular we show a connection between graphs maximizing the number of cycles with length matching their odd girth and Moore graphs. Continuing with strongly regular graphs we present a classificational result for strongly regular graphs. The approach is based on the so called star complement technique developed by Cvetković and Rowlinson.
Keywords: adjacency matrix; chromatic polynomials; convex cycle; Disertacije; konveksni cikel; krepko regularni grafi; kromatični polinomi; matrika sosednosti; strongly regular graphs; Teorija grafov; Tutte polynomial; Tuttov polinom