On the minimum real roots of the σ-polynomials and chromatic uniqueness of graphs

Abstract

Let ÿ(G) denote the minimum real root of the -polynomial of the complement of a graph G and (G) the minimum degree of G. In this paper, we give a characterization of all connected graphs G with ÿ(G) ¿ − 4. Using these results, we establish a su cient and necessary condition for a graph G with p vertices and (G) ¿ p − 3, to be chromatically unique. Many previously known results are generalized. As a byproduct, a problem of Du (Discrete Math. 162 (1996) 109-125) and a conjecture of Liu (Discrete Math. 172 (1997) 85-92) are conÿrmed.

On problems and conjectures on adjointly equivalent graphs

“…, G k . Then (iii) For n 4, From Lemmas 2.4 and 2.5 or [8][9][10], one can check that each pair of the graphs in R 1 defined below is adjointly equivalent. In what follows, we call it an adjointly equivalent transform if we use one of the graph in a pair of adjointly equivalent graphs to substitute for the other.…”

“…If e = uv is not an edge in any triangle of a graph G, thenh(G) = h(G − e) + xh(G − {u, v}).Lemma 2.6 (Ye and Li[8], Zhao et al[9,10]). (i) For n 1 and m 4, (h 1 (C m ), h 1 (P 2n )) = 1;(ii) For n 1 3 and n 2 4, h 1 (P n 1 )h 1 (C n 2 ) = h 1 (P n 1 +n 2 ) if and only if n 2 = n 1 + 1.…”

The chromatic equivalence classes of the complements of graphs with the minimum real roots of their adjoint polynomials greater than -4

“…In [13], Zhao, et al, gave a characterization of all connected graphs G with (G) − 4. In this paper, we determine [G] P for each graph G with (G) > − 4.…”

On the minimum real roots of the adjoint polynomial of a graph

“…By applying some properties of adjoint polynomials of graphs, many authors found some chromatically unique graphs, see [6,12,10,8,13,14,16,20,18,19]. In particular, by using some results of minimum real roots of adjoint polynomials of graphs, the authors in [20] gave a necessary and sufficient condition for G with the minimum degree δ(G) ≥ p(G)−3 to be χ-unique and the authors in [18] solved a problem on chromaticity of complete multipartite graphs posed by Koh and Teo in Graph and Combin.…”

“…We denote by β(G) the minimum real root of h(G, x). The authors in [20] determined all the connected graphs with β(G) ≥ −4. …”