Extremal regular graphs: independent sets and graph homomorphisms
TLDR
In this paper, a survey of extremal regular graphs with respect to the number of independent sets and graph homomorphisms is presented, in particular in the family of d-regular graphs.Abstract:
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 of of d-regular graphs, ...read more
Citations
More filters
Posted Content
Diagonal Ramsey via effective quasirandomness
Ashwin Sah,Toshihiro Shimada +1 more
TL;DR: In this article, the upper bound for diagonal Ramsey numbers was improved to R(k+1,k + 1,k+ 1 )expexpexp(c(log k)^2 )binom{2k}{k} for k ≥ 3.
Journal ArticleDOI
The number of independent sets in an irregular graph
TL;DR: Borders on the weighted versions of these quantities, i.e., the independent set polynomial, or equivalently the partition function of the hard-core model with a given fugacity on a graph, are proved.
Journal ArticleDOI
Counting independent sets in cubic graphs of given girth
Guillem Perarnau,Will Perkins +1 more
TL;DR: It is proved that in fact all Moore graphs are extremal for the scaled number of independent sets in regular graphs of a given minimum girth, maximizing this quantity if their girth is even and minimizing if odd.
Journal ArticleDOI
Extremes of the internal energy of the Potts model on cubic graphs
TL;DR: In this paper, the authors proved tight upper and lower bounds on the internal energy per particle (expected number of monochromatic edges per vertex) in the anti-ferromagnetic Potts model on cubic graphs at every temperature and for all $q \ge 2$.
Posted Content
Extremal regular graphs: the case of the infinite regular tree
TL;DR: In this article, it was shown that in many instances the infimum is not achieved by a finite graph, but a sequence of graphs with girth tending to infinity, in other words, the optimization problem is solved by the infinite $d$--regular tree.