Dong Fengming

Let G be a connected graph with vertex set V (G). A set S of vertices in G is called a weakly connected dominating set of G if (i) S is a dominating set of G and (ii) the graph obtained from G by removing all edges joining two vertices in V (G) \ S is connected. A weakly connected dominating set S of G is said to be minimum or a γw-set if |S| is minimum among all weakly connected dominating sets of G. We say that G is γw-unique if it has a unique γw-set. Recently, a constructive characterisation of γw-unique trees was obtained by Lemanska and Raczek [Czechoslovak Math. J. 59 (134) (2009), 95–100]. A graph is said to be cycle-disjoint if no two cycles in G have a vertex in common. In this paper, we extend the above result on trees by establishing a constructive characterisation of γw-unique cycle-disjoint graphs.

The chromatic polynomial of a simple graph G with n > 0 vertices is a polynomial Σnk =1α(G, k)(x)k of degree n, where (x)k = x(x−1) . . . (x−k+1) and α(G, k) is real for all k. The adjoint polynomial of G is defined to be Σnk=1α(G, k)μk, where G is the complement of G. We find the zeros of the adjoint polynomials of paths and cycles.

For any graph G, let W(G) be the set of vertices in G of degrees larger than 3. We show that for any bridgeless graph G, if W(G) is dominated by some component of G-W(G), then F(G,λ ) has no roots in (1; 2), where F(G,λ ) is the flow polynomial of G. This result generalizes the known result that F(G,λ ) has no roots in (1, 2) whenever |W(G)| ≤2. We also give some constructions to generate graphs whose flow polynomials have no roots in (1, 2).

Let (a1, a2, · · · , ak) denote the graph obtained by connecting two distinct vertices with k independent paths of lengths a1, a2, · · · , ak respectively. Assume that 2 ≤ a1 ≤ a2 ≤ · · · ≤ ak. We prove that the graph θ (a1, a2, · · · , ak) is chromatically unique if ak < a1 +a2, and find examples showing that θ (a1, a2, · · · , ak) may not be chromatically unique if ak = a1 + a2.

Let P be an edge-property of graphs. For any graph G we construct a polynomial Ψ(G, η,P), in an indeterminate η, in which the coefficient of ηr for any r ≥ 0 gives the number of subsets of E(G) that have cardinality r and satisfy P. An example is the well known matching polynomial of a graph. After studying the properties of Ψ(G, η,P) in general, we specialise to two particular edge-properties: that of being an edge-covering and that of inducing an acyclic subgraph. The resulting polynomials, called the edge-cover and acyclic polynomials respectively, are studied and recursive formulae for computing them are derived. As examples we calculate these polynomials for paths and cycles.

For any positive integers a, b, c, d, let Ra,b,c,d be the graph obtained from the complete graphs Ka,Kb,Kc and Kd by adding edges joining every vertex in Ka and Kc to every vertex in Kb and Kd. This paper shows that for arbitrary positive integers a, b, c and d, every root of the chromatic polynomial of Ra,b,c,d is either a real number or a non-real number with its real part equal to (a + b + c + d − 1)/2.

The chromatic polynomial 𝑃(𝐺,𝑥) of a graph 𝐺 of order 𝑛 can be expressed as ∑𝑛𝑖=1(−1)𝑛−𝑖𝑎𝑖𝑥𝑖 , where 𝑎𝑖 is interpreted as the number of broken‐cycle‐free spanning subgraphs of 𝐺 with exactly 𝑖 components. The parameter 𝜖(𝐺)=∑𝑛𝑖=1(𝑛−𝑖)𝑎𝑖/∑𝑛𝑖=1𝑎𝑖 is the mean size of a broken‐cycle‐free spanning subgraph of 𝐺 . In this article, we confirm and strengthen a conjecture proposed by Lundow and Markström in 2006 that 𝜖(𝑇𝑛)<𝜖(𝐺)<𝜖(𝐾𝑛) holds for any connected graph 𝐺 of order 𝑛 which is neither the complete graph 𝐾𝑛 nor a tree 𝑇𝑛 of order 𝑛 . The most crucial step of our proof is to obtain the interpretation of all 𝑎𝑖 's by the number of acyclic orientations of 𝐺 .

For a given graph G, let P (G , A) be the chromatic polynomial of G, where A is considered to be a real number. In this paper, we study the bounds for P (G , A )/P (G , A — 1) and P (G , A )/P (G - x, A), where x is a vertex in G, A > n and n is the number of vertices of G.