Preface For a century, one of the most famous problems in mathematics was to prove the four-colour theorem. This has spawned the development of many useful tools for solving graph colouring problems. In a paper in 1912, Birkhoff proposed a way of tackling the four-colour problem by introducing a function P (M, λ), defined for all positive integers λ, to be… (More)
The chromatic polynomial of a simple graph G with n > 0 vertices is a polynomial Σ n k=1 α(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 Σ n k=1 α(G, k)µ k , where G is the complement of G. We find the zeros of the adjoint polynomials of paths and cycles.
Two graphs are defined to be adjointly equivalent if their complements are chromatically equivalent. We study the properties of two invariants under adjoint equivalence.
Let G = (V, E) be a 2-connected plane graph on n vertices with outer face C such that every 2-vertex cut of G contains at least one vertex of C. Let P G (q) denote the chromatic polynomial of G. We show that (−1) n P G (q) > 0 for all 1 < q ≤ 1.2040.... This result is a corollary of a more general result that (−1) n Z G (q, w) > 0 for all 1 < q ≤ 1.2040...,… (More)
For a connected graph G and any non-empty S ⊆ V (G), S is called a weakly connected dominating set of G if the subgraph obtained from G by removing all edges each joining any two vertices in V (G) \ S is connected. The weakly connected domination number γ w (G) is defined to be the minimum integer k with |S| = k for some weakly connected dominating set S of… (More)
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… (More)
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 γ w-set if |S| is minimum… (More)
For any positive integers a, b, c, d, let R a,b,c,d be the graph obtained from the complete graphs K a , K b , K c and K d by adding edges joining every vertex in K a and K c to every vertex in K b and K d. This paper shows that for arbitrary positive integers a, b, c and d, every root of the chromatic polynomial of R a,b,c,d is either a real number or a… (More)