Embedability between right-angled Artin groups

In this article we study the right-angled Artin subgroups of a given right-angled Artin group. Starting with a graph \gam, we produce a new graph through a purely combinatorial procedure, and call it the extension graph \gam^e of \gam. We produce a second graph \gam^e_k, the clique graph of \gam^e, by adding extra vertices for each complete subgraph of \gam^e. We prove that each finite induced subgraph $\Lambda$ of \gam^e gives rise to an inclusion A(\Lambda)\to A(\gam). Conversely, we show that if there is an inclusion A(\Lambda)\to A(\gam) then $\Lambda$ is an induced subgraph of \gam^e_k. These results have a number of corollaries. Let $P_4$ denote the path on four vertices and let $C_n$ denote the cycle of length $n$. We prove that $A(P_4)$ embeds in A(\gam) if and only if $P_4$ is an induced subgraph of \gam. We prove that if $F$ is any finite forest then $A(F)$ embeds in $A(P_4)$. We recover the first author's result on co--contraction of graphs and prove that if \gam has no triangles and A(\gam) contains a copy of $A(C_n)$ for some $n\geq 5$, then \gam contains a copy of $C_m$ for some $5\le m\le n$. We also recover Kambites' Theorem, which asserts that if $A(C_4)$ embeds in A(\gam) then \gam contains an induced square. Finally, we determine precisely when there is an inclusion $A(C_m)\to A(C_n)$ and show that there is no "universal" two--dimensional right-angled Artin group.Comment: 35 pages. Added an appendix and a proof that the extension graph is quasi-isometric to a tre

Quantum Query Complexity of Subgraph Isomorphism and Homomorphism

Let $H$ be a fixed graph on $n$ vertices. Let $f_H(G) = 1$ iff the input graph $G$ on $n$ vertices contains $H$ as a (not necessarily induced) subgraph. Let $\alpha_H$ denote the cardinality of a maximum independent set of $H$. In this paper we show: $$Q(f_H) = \Omega\left(\sqrt{\alpha_H \cdot n}\right),$$ where $Q(f_H)$ denotes the quantum query complexity of $f_H$. As a consequence we obtain a lower bounds for $Q(f_H)$ in terms of several other parameters of $H$ such as the average degree, minimum vertex cover, chromatic number, and the critical probability. We also use the above bound to show that $Q(f_H) = \Omega(n^{3/4})$ for any $H$, improving on the previously best known bound of $\Omega(n^{2/3})$. Until very recently, it was believed that the quantum query complexity is at least square root of the randomized one. Our $\Omega(n^{3/4})$ bound for $Q(f_H)$ matches the square root of the current best known bound for the randomized query complexity of $f_H$, which is $\Omega(n^{3/2})$ due to Gr\"oger. Interestingly, the randomized bound of $\Omega(\alpha_H \cdot n)$ for $f_H$ still remains open. We also study the Subgraph Homomorphism Problem, denoted by $f_{[H]}$, and show that $Q(f_{[H]}) = \Omega(n)$. Finally we extend our results to the $3$-uniform hypergraphs. In particular, we show an $\Omega(n^{4/5})$ bound for quantum query complexity of the Subgraph Isomorphism, improving on the previously known $\Omega(n^{3/4})$ bound. For the Subgraph Homomorphism, we obtain an $\Omega(n^{3/2})$ bound for the same.Comment: 16 pages, 2 figure

A tool for filtering information in complex systems

We introduce a technique to filter out complex data-sets by extracting a subgraph of representative links. Such a filtering can be tuned up to any desired level by controlling the genus of the resulting graph. We show that this technique is especially suitable for correlation based graphs giving filtered graphs which preserve the hierarchical organization of the minimum spanning tree but containing a larger amount of information in their internal structure. In particular in the case of planar filtered graphs (genus equal to 0) triangular loops and 4 element cliques are formed. The application of this filtering procedure to 100 stocks in the USA equity markets shows that such loops and cliques have important and significant relations with the market structure and properties.Comment: 8 pages, 3 figures, 4 table