色多项式系数的几个结果

Some Results on the Coefficients of Chromatic Polynomials

The unimodal conjecture for chromatic polynomials was proposed by R. C. Read in1968. Few satisfactory results of this problem have so far been known:except: some spe-cial cases. In the present paper we introduce the notions of chromatic coefficient vectorsand glued graphs, and then give the extended result of Heron. Taking advantage of theconception of average edge numbers of glued graphs, we change the unimodal conjectureto the conjecture where the sequence of average edge numbers of glued graphs of allorders is a monotone decreasing, sequence and it is proved to be true for forests andcomplete graphs, In the paper we also suggest the notion of generalized trees and givea necessary and sufficient condition for generalized trees, Then we prove that for gener-alized trees, the unimodal conjecture is true.