Index theorems for quantum graphs

Splitter Theorems for Cubic Graphs

Let k g be the class of k connected cubic graphs of girth at least g For several choices of k and g we determine a set Ok g of graph operations for which if G and H are graphs in k g G H and G contains H topologically then some operation in Ok g can be applied to G to result in a smaller graph G in k g such that on one hand G is contained in G topologically and on the other hand G contains H to...

The second geometric-arithmetic index for trees and unicyclic graphs

Let $G$ be a finite and simple graph with edge set $E(G)$. The second geometric-arithmetic index is defined as $GA_2(G)=sum_{uvin E(G)}frac{2sqrt{n_un_v}}{n_u+n_v}$, where $n_u$ denotes the number of vertices in $G$ lying closer to $u$ than to $v$. In this paper we find a sharp upper bound for $GA_2(T)$, where $T$ is tree, in terms of the order and maximum degree o...

Nordhaus-Gaddum type results for the Harary index of graphs

The emph{Harary index} $H(G)$ of a connected graph $G$ is defined as $H(G)=sum_{u,vin V(G)}frac{1}{d_G(u,v)}$ where $d_G(u,v)$ is the distance between vertices $u$ and $v$ of $G$. The Steiner distance in a graph, introduced by Chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. For a connected graph $G$ of order at least $2$ ...

Convergence theorems for quantum annealing

Abstract. We prove several theorems to give sufficient conditions for convergence of quantum annealing, which is a protocol to solve generic optimization problems by quantum dynamics. In particular the property of strong ergodicity is proved for the path-integral Monte Carlo implementation of quantum annealing for the transverse Ising model under a power decay of the transverse field. This resu...

Limit theorems for quantum entropies