This survey concerns regular graphs that are extremal with respect to the number of
independent sets and, more generally, graph homomorphisms. More precisely, in the family …

Let H be a graph allowing loops as well as vertex and edge weights. We prove that, for every
triangle-free graph G without isolated vertices, the weighted number of graph …

We prove a tight upper bound on the independence polynomial (and total number of
independent sets) of cubic graphs of girth at least 5. The bound is achieved by unions of the …

Abstract Settling Kahn's conjecture (2001), we prove the following upper bound on the
number i (G) of independent sets in a graph G without isolated vertices: i (G)≤∏ uv∈ E (G) i …

In 1971, Tomescu conjectured that every connected graph G on n vertices with chromatic
number k≥ 4 has at most k!(k− 1) n− k proper k-colorings. Recently, Knox and Mohar proved …

In this thesis we explore instances in which tools from continuous optimisation can be used
to solve problems in extremal graph and hypergraph theory. We begin by introducing a …

In this paper we study the following problem. Let $ A $ be a fixed graph, and let $\hom (G, A)
$ denote the number of homomorphisms from a graph $ G $ to $ A $. Furthermore, let $ v (G) …

We give tight upper and lower bounds on the internal energy per particle in the
antiferromagnetic $ q $-state Potts model on $4 $-regular graphs, for $ q\ge 5$. This proves …

Partition functions arise in statistical physics and probability theory as the normalizing
constant of Gibbs measures and in combinatorics and graph theory as graph polynomials …

We consider several extremal problems of maximizing the number of colorings and
independent sets in some graph families with fixed chromatic number and order. First, we …