Max k-cut and the smallest eigenvalue

Let $G$ be a graph of order $n$ and size $m$, and let $\mathrm{mc}_{k}\left( G\right)$ be the maximum size of a $k$-cut of $G.$ It is shown that $$\mathrm{mc}_{k}\left( G\right) \leq\frac{k-1}{k}\left( m-\frac{\mu_{\min }\left( G\right) n}{2}\right) ,$$ where $\mu_{\min}\left( G\right)$ is the smallest eigenvalue of the adjacency matrix of $G.$ An infinite class of graphs forcing equality in this bound is constructed.Comment: 5 pages. Some typos corrected in v

Special Section on the Forty-First Annual ACM Symposium on Theory of Computing (STOC 2009)

This issue of SICOMP contains nine specially selected papers from the Forty-first Annual ACM Symposium on the Theory of Computing, otherwise known as STOC 2009, held May 31 to June 2 in Bethesda, Maryland. The papers here were chosen to represent both the excellence and the broad range of the STOC program. The papers have been revised and extended by the authors, and subjected to the standard thorough reviewing process of SICOMP. The program committee consisted of Susanne Albers, Andris Ambainis, Nikhil Bansal, Paul Beame, Andrej Bogdanov, Ran Canetti, David Eppstein, Dmitry Gavinsky, Shafi Goldwasser, Nicole Immorlica, Anna Karlin, Jonathan Katz, Jonathan Kelner, Subhash Khot, Ravi Kumar, Leslie Ann Goldberg, Michael Mitzenmacher (Chair), Kamesh Munagala, Rasmus Pagh, Anup Rao, Rocco Servedio, Mikkel Thorup, Chris Umans, and Lisa Zhang. They accepted 77 papers out of 321 submissions

Detecting and Characterizing Small Dense Bipartite-like Subgraphs by the Bipartiteness Ratio Measure

We study the problem of finding and characterizing subgraphs with small \textit{bipartiteness ratio}. We give a bicriteria approximation algorithm \verb|SwpDB| such that if there exists a subset $S$ of volume at most $k$ and bipartiteness ratio $\theta$, then for any $0<\epsilon<1/2$, it finds a set $S'$ of volume at most $2k^{1+\epsilon}$ and bipartiteness ratio at most $4\sqrt{\theta/\epsilon}$. By combining a truncation operation, we give a local algorithm \verb|LocDB|, which has asymptotically the same approximation guarantee as the algorithm \verb|SwpDB| on both the volume and bipartiteness ratio of the output set, and runs in time $O(\epsilon^2\theta^{-2}k^{1+\epsilon}\ln^3k)$, independent of the size of the graph. Finally, we give a spectral characterization of the small dense bipartite-like subgraphs by using the $k$th \textit{largest} eigenvalue of the Laplacian of the graph.Comment: 17 pages; ISAAC 201

Network Density of States

Spectral analysis connects graph structure to the eigenvalues and eigenvectors of associated matrices. Much of spectral graph theory descends directly from spectral geometry, the study of differentiable manifolds through the spectra of associated differential operators. But the translation from spectral geometry to spectral graph theory has largely focused on results involving only a few extreme eigenvalues and their associated eigenvalues. Unlike in geometry, the study of graphs through the overall distribution of eigenvalues - the spectral density - is largely limited to simple random graph models. The interior of the spectrum of real-world graphs remains largely unexplored, difficult to compute and to interpret. In this paper, we delve into the heart of spectral densities of real-world graphs. We borrow tools developed in condensed matter physics, and add novel adaptations to handle the spectral signatures of common graph motifs. The resulting methods are highly efficient, as we illustrate by computing spectral densities for graphs with over a billion edges on a single compute node. Beyond providing visually compelling fingerprints of graphs, we show how the estimation of spectral densities facilitates the computation of many common centrality measures, and use spectral densities to estimate meaningful information about graph structure that cannot be inferred from the extremal eigenpairs alone.Comment: 10 pages, 7 figure