An algebraic analysis of the graph modularity

One of the most relevant tasks in network analysis is the detection of community structures, or clustering. Most popular techniques for community detection are based on the maximization of a quality function called modularity, which in turn is based upon particular quadratic forms associated to a real symmetric modularity matrix $M$, defined in terms of the adjacency matrix and a rank one null model matrix. That matrix could be posed inside the set of relevant matrices involved in graph theory, alongside adjacency, incidence and Laplacian matrices. This is the reason we propose a graph analysis based on the algebraic and spectral properties of such matrix. In particular, we propose a nodal domain theorem for the eigenvectors of $M$; we point out several relations occurring between graph's communities and nonnegative eigenvalues of $M$; and we derive a Cheeger-type inequality for the graph optimal modularity

P?=NP as minimization of degree 4 polynomial, integration or Grassmann number problem, and new graph isomorphism problem approaches

While the P vs NP problem is mainly approached form the point of view of discrete mathematics, this paper proposes reformulations into the field of abstract algebra, geometry, fourier analysis and of continuous global optimization - which advanced tools might bring new perspectives and approaches for this question. The first one is equivalence of satisfaction of 3-SAT problem with the question of reaching zero of a nonnegative degree 4 multivariate polynomial (sum of squares), what could be tested from the perspective of algebra by using discriminant. It could be also approached as a continuous global optimization problem inside $[0,1]^n$, for example in physical realizations like adiabatic quantum computers. However, the number of local minima usually grows exponentially. Reducing to degree 2 polynomial plus constraints of being in $\{0,1\}^n$, we get geometric formulations as the question if plane or sphere intersects with $\{0,1\}^n$. There will be also presented some non-standard perspectives for the Subset-Sum, like through convergence of a series, or zeroing of $\int_0^{2\pi} \prod_i \cos(\varphi k_i) d\varphi$ fourier-type integral for some natural $k_i$. The last discussed approach is using anti-commuting Grassmann numbers $\theta_i$, making $(A \cdot \textrm{diag}(\theta_i))^n$ nonzero only if $A$ has a Hamilton cycle. Hence, the P$\ne$NP assumption implies exponential growth of matrix representation of Grassmann numbers. There will be also discussed a looking promising algebraic/geometric approach to the graph isomorphism problem -- tested to successfully distinguish strongly regular graphs with up to 29 vertices.Comment: 19 pages, 8 figure

Package of facts and theorems for efficiently generating entanglement criteria for many qubits

We present a package of mathematical theorems, which allow to construct multipartite entanglement criteria. Importantly, establishing bounds for certain classes of entanglement does not take an optimization over continuous sets of states. These bonds are found from the properties of commutativity graphs of operators used in the criterion. We present two examples of criteria constructed according to our method. One of them detects genuine 5-qubit entanglement without ever referring to correlations between all five qubits.Comment: 5 pages, 4 figure