B\mathcal{B}-partitions, application to determinant and permanent of graphs

Let $G$ be a graph(directed or undirected) having $k$ number of blocks. A $\mathcal{B}$-partition of $G$ is a partition into $k$ vertex-disjoint subgraph (\hat{B_1},\hat{B_1},\hdots,\hat{B_k}) such that $\hat{B}_i$ is induced subgraph of $B_i$ for i=1,2,\hdots,k. The terms $\prod_{i=1}^{k}\det(\hat{B}_i),\ \prod_{i=1}^{k}\text{per}(\hat{B}_i)$ are det-summands and per-summands, respectively, corresponding to the $\mathcal{B}$-partition. The determinant and permanent of a graph having no loops on its cut-vertices is equal to summation of det-summands and per-summands, respectively, corresponding to all possible $\mathcal{B}$-partitions. Thus, in this paper we calculate determinant and permanent of some graphs, which include block graph with negatives cliques, signed unicyclic graph, mix complete graph, negative mix complete graph, and star mix block graphs

Algorithm for B\mathcal{B}-partitions, parameterized complexity of the matrix determinant and permanent

Every square matrix $A=(a_{uv})\in \mathcal{C}^{n\times n}$ can be represented as a digraph having $n$ vertices. In the digraph, a block (or 2-connected component) is a maximally connected subdigraph that has no cut-vertex. The determinant and the permanent of a matrix can be calculated in terms of the determinant and the permanent of some specific induced subdigraphs of the blocks in the digraph. Interestingly, these induced subdigraphs are vertex-disjoint and they partition the digraph. Such partitions of the digraph are called the $\mathcal{B}$-partitions. In this paper, first, we develop an algorithm to find the $\mathcal{B}$-partitions. Next, we analyze the parameterized complexity of matrix determinant and permanent, where, the parameters are the sizes of blocks and the number of cut-vertices of the digraph. We give a class of combinations of cut-vertices and block sizes for which the parametrized complexities beat the state of art complexities of the determinant and the permanent.Comment: arXiv admin note: text overlap with arXiv:1701.0442

A Complex Network Approach for Collaborative Recommendation

Collaborative filtering (CF) is the most widely used and successful approach for personalized service recommendations. Among the collaborative recommendation approaches, neighborhood based approaches enjoy a huge amount of popularity, due to their simplicity, justifiability, efficiency and stability. Neighborhood based collaborative filtering approach finds K nearest neighbors to an active user or K most similar rated items to the target item for recommendation. Traditional similarity measures use ratings of co-rated items to find similarity between a pair of users. Therefore, traditional similarity measures cannot compute effective neighbors in sparse dataset. In this paper, we propose a two-phase approach, which generates user-user and item-item networks using traditional similarity measures in the first phase. In the second phase, two hybrid approaches HB1, HB2, which utilize structural similarity of both the network for finding K nearest neighbors and K most similar items to a target items are introduced. To show effectiveness of the measures, we compared performances of neighborhood based CFs using state-of-the-art similarity measures with our proposed structural similarity measures based CFs. Recommendation results on a set of real data show that proposed measures based CFs outperform existing measures based CFs in various evaluation metrics.Comment: 22 Page

Linear time algorithm to check the singularity of block graphs

A block graph is a graph in which every block is a complete graph. Let $G$ be a block graph and let $A(G)$ be its (0,1)-adjacency matrix. Graph $G$ is called nonsingular (singular) if $A(G)$ is nonsingular (singular). Characterizing nonsingular block graphs is an interesting open problem proposed by Bapat and Roy in 2013. In this article, we give a linear time algorithm to check whether a given block graph is singular or not

Isomorphism of Skew-Holomorphic Harmonic Maass-Jacobi Forms and Certain Weak Harmonic Maass Forms

Recently Bringmann, Raum and Richter generalised the definition of Jacobi forms and Skoruppa's skew-holomorphic Jacobi forms by intertwining with harmonic Maass forms. We prove the isomorphism of the Kohnen's plus space analogue of harmonic Maass forms of weight $k-1/2$ for $\Gamma_0(4m)$ and the space of these skew-holomorphic harmonic Maass-Jacobi forms of weight $k$ and index $m$ for $k$ odd and $m=1$ or a prime.Comment: 9 page

On the classical dynamics of charged particle in special class of spatially non-uniform magnetic field

Motion of a charged particle in uniform magnetic field has been studied in detail, classically as well as quantum mechanically. However, classical dynamics of a charged particle in non-uniform magnetic field is solvable only for some specific cases. We present in this paper, a general integral equation for some specific class of non-uniform magnetic field and its solutions for some of them. We also examine the supersymmetry of Hamiltonians in exponentially decaying magnetic field with radial dependence and conclude that this kind of non-uniformity breaks supersymmetry.Comment: 15 pages, 5 figures, Accepted for publication in Indian journal of physics, Springe

Mock Eisenstein Series

In this paper, we give explicit construction of harmonic weak Maass forms which are pullbacks of the general Eisenstein series of integral and half-integral weight under the shadow operator. Using our construction, we show that the space of harmonic weak Maass forms which have atmost polynomial growth at every cusp is finite dimensional and is spanned by the preimages of Eisenstein series. As an application, we construct the preimages of $\Theta^k$ where $\Theta$ is the classical Jacobi theta function for $k\in\{3,5,7\}$ and show that the preimages are Hecke eigenforms. In the integral case, we recover some results of Herrero and Pippich (2018) and in half-integral case, we recover the results of Rhoades and Waldherr (2011).Comment: 20 pages, corrected some typos, added new results and a sectio

Nonsingular (Vertex-Weighted) Block Graphs

A graph $G$ is \emph{nonsingular (singular)} if its adjacency matrix $A(G)$ is nonsingular (singular). In this article, we consider the nonsingularity of block graphs, i.e., graphs in which every block is a clique. Extending the problem, we characterize nonsingular vertex-weighted block graphs in terms of reduced vertex-weighted graphs resulting after successive deletion and contraction of pendant blocks. Special cases where nonsingularity of block graphs may be directly determined are discussed

Complete multipartite graphs that are determined, up to switching, by their Seidel spectrum

It is known that complete multipartite graphs are determined by their distance spectrum but not by their adjacency spectrum. The Seidel spectrum of a graph $G$ on more than one vertex does not determine the graph, since any graph obtained from $G$ by Seidel switching has the same Seidel spectrum. We consider $G$ to be determined by its Seidel spectrum, up to switching, if any graph with the same spectrum is switching equivalent to a graph isomorphic to $G$. It is shown that any graph which has the same spectrum as a complete $k$-partite graph is switching equivalent to a complete $k$-partite graph, and if the different partition sets sizes are $p_1,\ldots, p_l$, and there are at least three partition sets of each size $p_i$, $i=1,\ldots, l$, then $G$ is determined, up to switching, by its Seidel spectrum. Sufficient conditions for a complete tripartite graph to be determined by its Seidel spectrum are discussed, and a conjecture is made on complete tripartite graphs on more than 18 vertices.Comment: 12 page 2 figure

Nonsingular Block Graphs: An Open Problem

A block graph is a graph in which every block is a complete graph. Let $G$ be a block graph and let $A(G)$ be its (0,1)-adjacency matrix. Graph $G$ is called nonsingular (singular) if $A(G)$ is nonsingular (singular). An interesting open problem, proposed in 2013 by Bapat and Roy, is to characterize nonsingular block graphs. In this article, we present some classes of nonsingular and singular block graphs and related conjectures.Comment: In this draft the Conjectures are wron