It is shown that the general 8-vertex model on the honeycomb lattice is always reducible to an Ising model in a nonzero but generally complex magnetic field. In the most general case of the staggered 8-vertex model characterized by 16 independent vertex weights, the equivalent Ising model has three anisotropic interactions and a staggered magnetic field which assumes two different values on the two sublattices.
The Merrifield–Simmons index
i
G
of a graph
G
is defined as the number of subsets of the vertex set, in which any two vertices are nonadjacent, i.e., the number of independent vertex sets of
G
. In this paper, we determine the minimum Merrifield–Simmons index of unicyclic graphs with
n
vertices and diameter at most four.
The Merrifield-Simmons indexi(G)of a graphGis defined as the number of subsets of the vertex set, in which any two vertices are nonadjacent, that is, the number of independent vertex sets ofGThe Hosoya indexz(G)of a graphGis defined as the total number of independent edge subsets, that is, the total number of its matchings. ByC(n,k,λ)we denote the set of graphs withnvertices,kcycles, the length of every cycle isλ, and all the edges not on the cycles are pendant edges which are attached to the same vertex. In this paper, we investigate the Merrifield-Simmons indexi(G)and the Hosoya indexz(G)for a graphGinC(n,k,λ).
This paper is devoted to the proof of the surprising fact that almost all trees have an even number of independent vertex subsets (in the sense that the proportion of those trees with an odd number of independent sets tends to $0$ as the number of vertices approaches $\infty$) and to its generalisation to other moduli: for fixed $m$, the probability that a randomly chosen tree on $n$ vertices has a number of independent subsets that is divisible by $m$ tends to $1$ as $n \to \infty$.
Fast community structure local uncovering by independent vertex-centred process
