On the roots of $\sigma$-polynomials

SIAM Journal on Discrete Mathematics, 2010

For a family S of graphs, let ω(S) be the supremum of t : 1 < t ≤ 2 such that P (G, λ) = 0 for all G ∈ S and all λ ∈ (1, t). In this paper we show that ω(S) = ω(S ∩ K) for any family S of graphs satisfying certain conditions, where K is a special family of graphs. This result makes it much easier to determine ω(S) for such families S.

2015

The chromatic polynomial, PΓ(x) of a graph Γ, is a polynomial that when evaluated at a positive integer k, is the number of proper k colorings of the graph Γ. We can then find the orbital chromatic polynomial OPΓ,G(x) of a graph Γ and a group G of automorphisms of Γ, which is a polynomial whose value at a positive integer k is the number of orbits of k-colorings of a graph Γwhen acted upon by the group G. By considering the roots of the orbital chromatic and chromatic polynomials, the similarities and differences of these polynomials is studied. Specifically we work toward proving a conjecture concerning the gap between the real roots of the chromatic polynomial and the real roots of the orbital chromatic polynomial.

In this note we consider a finite graph without loops and multiple edges. The colouring of a graph G in A colours is the colouring of its vertices in such a way that no two of adjacent vertices have the same colours and the number of used colours does not exceed A [1,4]. Two colourings of graph G are called different if there exists at least one vertex which changes colour when passing from one colouring to another. If F n is a full (also known as 'complete') graph on n vertices, & < n , then P(F k c F H , pi, A) = M (*>(A-*)<""*>, where M <*> = M (M-l)(/i-2). .. (/ *-* + 1). In particular P(F t <= F,,, A, A) = A W (A-*)<"-*> = A (n). If £ " is empty (also known as 'null' or 'totally disconnected' graph on n vertices) then P(E k a E, n fi, A) = fi k \"-k , in particular P(E k c £ " , A, A) = A". Let G = (X,V), \X\ = n be a graph and G o = (* " , K o), X 0 ^X 0 \X 0 \ = m be an induced subgraph of G. Let also x,y <= Xbe two non-adjacent vertices in G. We construct the graph G, from G by joining x and _y by an edge and the graph G 2 , obtained from G by contracting x and y into single vertex. Then we can observe that the following equality is true: P(G 0 ^G, ti, A) = P(G l 0 ^G u fi,\) + P(Gl<=G 2 , /i,A), (1) where Gi = Go = G o if x,_y ^ A'o, and G o , Go are the subgraphs induced respectively by X ih X 0 U{x,y} otherwise. It is so because P (G 0 c G 2 , M , A) equals the number of colourings for which x and y have the same colour. If we perform this operation further as much as possible we obtain P(G 0 = G, n, A) = X P{F kl c Fn,, M, A). /=i Since P(F k <= F n ,fj.,A) = P(F' k <^F n ,p,A) for any two fc-vertex complete subgraphs F A ., F' k of F,,, we can write P(G 0 cz G, n, A) = 5 t o, y P(F y ^ F h M , A). In order to clear up the meaning of coefficients a, y we consider one more class of colourings. A colouring of a graph is called an /-colouring, if exactly / colours are used, but Glasgow Math. J. 36 (1994) 265-267.

2005

Discrete Mathematics & Theoretical Computer Science, 2003

Let P(G; x, y) be the number of vertex colorings φ : V → {1, ..., x} of an undirected graph G = (V, E) such that for all edges {u, v} ∈ E the relations φ(u) ≤ y and φ(v) ≤ y imply φ(u) = φ(v). We show that P(G; x, y) is a polynomial in x and y which is closely related to Stanley's chromatic symmetric function, and which simultaneously generalizes the chromatic polynomial, the independence polynomial, and the matching polynomial of G. We establish two general expressions for this new polynomial: one in terms of the broken circuit complex, and one in terms of the lattice of forbidden colorings. Finally, we give explicit expressions for the generalized chromatic polynomial of complete graphs, complete bipartite graphs, paths, and cycles, and show that P(G; x, y) can be evaluated in polynomial time for trees and graphs of restricted pathwidth.