The independence polynomial of a graph G is the polynomial ∑A x| A |, summed over all independent subsets A ⊆ V (G). We prove that if G is clawfree, then all the roots of its independence polynomial are real. This extends a theorem of Heilmann and Lieb [O.J. Heilmann, E.H. Lieb, Theory of monomer-dimer systems, Comm. Math. Phys. 25 (1972) 190-232], answering a question posed by Hamidoune [Y.O. Hamidoune, On the numbers of independent k-sets in a clawfree graph, J. Combin. Theory Ser. B 50 (1990) 241-244] and Stanley [R.P. Stanley, Graph colorings and related symmetric functions: Ideas and applications, Discrete Math. 193 (1998) 267-286].
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Clawfree graphs
- Independence polynomial