There are many ways of calculating a graphs characteristic polynomial; a lesser known method is a formula called the figure equation. The figure equation provides a direct link between a graph’s structure and the coefficients of its characteristic polynomial. This method does not use determinants, but calculates the characteristic polynomial of any graph by counting cycles. We give a complete combinatorial analysis of four increasingly complex graph families, which yields closed formulae for their characteristic polynomial. In this paper, we introduce the figure equation, prove formulae for the line, cyclic, ladder, and dihedral graphs, and examine connections among these graph families including isospectrality and graph covering maps.

Author Bio

Taylor Coon is a senior at the University of Rochester completing adouble degree in mathematics and music. This paper is the result ofresearch conducted at the 2006 REU, "Geometry and Physics on Graphs" atCanisius College under Terrance Bisson. Taylor hopes to attend graduateschool in mathematics.