Chromatic roots and limits of dense graphs

In this short note we observe that recent results of Abert and Hubai and of Csikvari and Frenkel about Benjamini--Schramm continuity of the holomorphic moments of the roots of the chromatic polynomial extend to the theory of dense graph sequences. We offer a number of problems and conjectures motivated by this observation.Comment: 9 page

Statistical matching theory

Non UBCUnreviewedAuthor affiliation: MITPostdoctora

Homomorphisms of Trees into a Path

Let hom(G,H) denote the number of homomorphisms from a graph G to a graph H. In this paper we study the number of homomorphisms of trees into a path, and prove that hom(P[subscript m],P[subscript n]) ≤ hom(T[subscript m],P[subscript n]) ≤ hom(S[subscript m],P[subscript n]), where T[subscript m] is any tree on m vertices, and P[subscript m] and S[subscript m] denote the path and star on m vertices, respectively. This completes the study of extremal problems concerning the number of homomorphisms between trees started in the paper Graph Homomorphisms Between Trees [Electron. J. Combin., 21 (2014), 4.9] written by the authors of the current paper

On the Number of Forests and Connected Spanning Subgraphs

Let F(G) be the number of forests of a graph G. Similarly let C(G) be the number of connected spanning subgraphs of a connected graph G. We bound F(G) and C(G) for regular graphs and for graphs with a fixed average degree. Among many other things we study f(d) = sup(G epsilon Gd) F(G)(1/nu)(G), where G(d) is the family of d-regular graphs, and upsilon(G) denotes the number of vertices of a graph G. We show that f(3) = 2(3/2), and if (G(n))(n) is a sequence of 3-regular graphs with the length of the shortest cycle tending to infinity, then lim(n ->infinity) F(G(n))(1/nu(Gn)) = 2(3/2). We also improve on the previous best bounds on f(d) for 4 <= d <= 9