Predicting inhibition of microsomal p-hydroxylation of aniline by aliphatic alcohols: A QSAR approach based on the weighted path numbers

Weighted path numbers are used to build QSAR models for predicting inhibition of microsomal p-hydroxylation of aniline by aliphatic alcohols. Models with two, three and four weighted path numbers are considered. Fit and cross-validated statistical parameters are used to measure the model quality. The best statistical parameters possess models with four weighted path numbers. Comparison with models from the literature favors models based on the weighted path numbers

Predicting Inhibition of Microsomal p-Hydroxylation of Aniline by Aliphatic Alcohols: A QSAR Approach Based on the Weighted Path Numbers

Weighted path numbers are used to build QSAR models for predicting inhibition of microsomal p-hydroxylation of aniline by aliphatic alcohols. Models with two, three and four weighted path numbers are considered. Fit and cross-validated statistical parameters are used to measure the model quality. The best statistical parameters possess models with four weighted path numbers. Comparison with models from the literature favors models based on the weighted path numbers

Limit theory of combinatorial optimization for random geometric graphs

In the random geometric graph $G(n,r_n)$, $n$ vertices are placed randomly in Euclidean $d$-space and edges are added between any pair of vertices distant at most $r_n$ from each other. We establish strong laws of large numbers (LLNs) for a large class of graph parameters, evaluated for $G(n,r_n)$ in the thermodynamic limit with $nr_n^d =$ const., and also in the dense limit with $n r_n^d \to \infty$, $r_n \to 0$. Examples include domination number, independence number, clique-covering number, eternal domination number and triangle packing number. The general theory is based on certain subadditivity and superadditivity properties, and also yields LLNs for other functionals such as the minimum weight for the travelling salesman, spanning tree, matching, bipartite matching and bipartite travelling salesman problems, for a general class of weight functions with at most polynomial growth of order $d-\varepsilon$, under thermodynamic scaling of the distance parameter.Comment: 64 page

The average cut-rank of graphs

The cut-rank of a set $X$ of vertices in a graph $G$ is defined as the rank of the $X \times (V(G)\setminus X)$ matrix over the binary field whose $(i,j)$-entry is $1$ if the vertex $i$ in $X$ is adjacent to the vertex $j$ in $V(G)\setminus X$ and $0$ otherwise. We introduce the graph parameter called the average cut-rank of a graph, defined as the expected value of the cut-rank of a random set of vertices. We show that this parameter does not increase when taking vertex-minors of graphs and a class of graphs has bounded average cut-rank if and only if it has bounded neighborhood diversity. This allows us to deduce that for each real $\alpha$, the list of induced-subgraph-minimal graphs having average cut-rank larger than (or at least) $\alpha$ is finite. We further refine this by providing an upper bound on the size of obstruction and a lower bound on the number of obstructions for average cut-rank at most (or smaller than) $\alpha$ for each real $\alpha\ge0$. Finally, we describe explicitly all graphs of average cut-rank at most $3/2$ and determine up to $3/2$ all possible values that can be realized as the average cut-rank of some graph.Comment: 22 pages, 1 figure. The bound $x_n$ is corrected. Accepted to European J. Combinatoric