A construction of pooling designs with surprisingly high degree of error correction

It is well-known that many famous pooling designs are constructed from mathematical structures by the "containment matrix" method. In this paper, we propose another method and obtain a family of pooling designs with surprisingly high degree of error correction based on a finite set. Given the numbers of items and pools, the error-tolerant property of our designs is much better than that of Macula's designs when the size of the set is large enough

Identifying codes of corona product graphs

For a vertex $x$ of a graph $G$, let $N_G[x]$ be the set of $x$ with all of its neighbors in $G$. A set $C$ of vertices is an {\em identifying code} of $G$ if the sets $N_G[x]\cap C$ are nonempty and distinct for all vertices $x$. If $G$ admits an identifying code, we say that $G$ is identifiable and denote by $\gamma^{ID}(G)$ the minimum cardinality of an identifying code of $G$. In this paper, we study the identifying code of the corona product $H\odot G$ of graphs $H$ and $G$. We first give a necessary and sufficient condition for the identifiable corona product $H\odot G$, and then express $\gamma^{ID}(H\odot G)$ in terms of $\gamma^{ID}(G)$ and the (total) domination number of $H$. Finally, we compute $\gamma^{ID}(H\odot G)$ for some special graphs $G$

On the metric dimension and fractional metric dimension for hierarchical product of graphs

A set of vertices $W$ {\em resolves} a graph $G$ if every vertex of $G$ is uniquely determined by its vector of distances to the vertices in $W$. The {\em metric dimension} for $G$, denoted by $\dim(G)$, is the minimum cardinality of a resolving set of $G$. In order to study the metric dimension for the hierarchical product $G_2^{u_2}\sqcap G_1^{u_1}$ of two rooted graphs $G_2^{u_2}$ and $G_1^{u_1}$, we first introduce a new parameter, the {\em rooted metric dimension} \rdim(G_1^{u_1}) for a rooted graph $G_1^{u_1}$. If $G_1$ is not a path with an end-vertex $u_1$, we show that \dim(G_2^{u_2}\sqcap G_1^{u_1})=|V(G_2)|\cdot\rdim(G_1^{u_1}), where $|V(G_2)|$ is the order of $G_2$. If $G_1$ is a path with an end-vertex $u_1$, we obtain some tight inequalities for $\dim(G_2^{u_2}\sqcap G_1^{u_1})$. Finally, we show that similar results hold for the fractional metric dimension.Comment: 11 page

The eigenvalues of qq-Kneser graphs

In this note, we prove some combinatorial identities and obtain a simple form of the eigenvalues of $q$-Kneser graphs

On finite groups all of whose cubic Cayley graphs are integral

For any positive integer $k$, let $\mathcal{G}_k$ denote the set of finite groups $G$ such that all Cayley graphs ${\rm Cay}(G,S)$ are integral whenever $|S|\le k$. Est${\rm \acute{e}}$lyi and Kov${\rm \acute{a}}$cs \cite{EK14} classified $\mathcal{G}_k$ for each $k\ge 4$. In this paper, we characterize the finite groups each of whose cubic Cayley graphs is integral. Moreover, the class $\mathcal{G}_3$ is characterized. As an application, the classification of $\mathcal{G}_k$ is obtained again, where $k\ge 4$.Comment: 11 pages, accepted by Journal of Algebra and its Applications on June 201

Pooling designs with surprisingly high degree of error correction in a finite vector space

Pooling designs are standard experimental tools in many biotechnical applications. It is well-known that all famous pooling designs are constructed from mathematical structures by the "containment matrix" method. In particular, Macula's designs (resp. Ngo and Du's designs) are constructed by the containment relation of subsets (resp. subspaces) in a finite set (resp. vector space). Recently, we generalized Macula's designs and obtained a family of pooling designs with more high degree of error correction by subsets in a finite set. In this paper, as a generalization of Ngo and Du's designs, we study the corresponding problems in a finite vector space and obtain a family of pooling designs with surprisingly high degree of error correction. Our designs and Ngo and Du's designs have the same number of items and pools, respectively, but the error-tolerant property is much better than that of Ngo and Du's designs, which was given by D'yachkov et al. \cite{DF}, when the dimension of the space is large enough