International Journal of Mathematics and Mathematical Sciences
Volume 8 (1985), Issue 4, Pages 697-705
On coefficients of circuit polynomials and characteristic polynomials
Department of Mathematics, The University of the West Indies, St. Augustine, Trinidad and Tobago
Received 10 May 1984
Copyright © 1985 E. J. Farrell. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Results are given from which expressions for the coefficients of the simple
circuit polynomial of a graph can be obtained in terms of subgraphs of the graph. From these are deduced parallel results for the coefficients of the characteristic polynomial of a graph. Some specific results are presented on the parities of the coefficients of characteristic polynomials. A characterization is then determined for graphs in which the number of sets of independent edges is always even. This leads to an interesting link between matching polynomials and characteristic polynomials. Finally explicit formulae are derived for the number of ways of covering two well known
families of graphs with node disjoint circuits, and for the first few coefficients of their characteristic polynomials.