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Anirban Banerjee

Publications and source records attributed to Anirban Banerjee.

At least 19 recordsLinked to original sources
arXiv2024

Spectral decomposition of hypergraph automorphism compatible matrices

This study explores the relationship between hypergraph automorphisms and the spectral properties of matrices associated with hypergraphs. For an automorphism $f$, an \( f \)-compatible matrices capture aspects of the symmetry, represented by \( f \), within the hypergraph. First, we explore rotation, a specific kind of automorphism and find that the spectrum of any matrix compatible with a rotation can be decomposed into the spectra of smaller matrices associated with that rotation. We show that the spectrum of any \(f\)-compatible matrix can be decomposed into the spectra of smaller matrices associated with the component rotations comprising \( f \). Further, we study a hypergraph symmetry termed unit-automorphism, which induces bijections on the hyperedges, though not necessarily on the vertex set. We show that unit automorphisms also lead to the spectral decomposition of compatible matrices.

math.CO↗
arXiv2023

Adjacency spectra of some subdivision hypergraphs

Here, we define a subdivision operation for a hypergraph and compute all the eigenvalues of the subdivision of regular and certain non-regular hypergraphs. In non-regular hypergraphs, we investigate the power of regular graphs, various types of hyperflowers, and the squid-like hypergraph. Using our subdivision operation, we also show how to construct non-regular non-isomorphic cospectral hypergraphs.

math.CO↗
arXiv2023

On the Spectra of Threshold Hypergraphs

Starting with an isolated vertex, here we construct a threshold hypergraph by repeatedly adding an isolated vertex or a $k$-dominating vertex set. We represent a threshold hypergraph by a string of non-negative integers and find the Laplacian spectrum of threshold hypergraphs from their string representation. We also compute the complete Laplacian spectrum of certain threshold hypergraphs from the Ferrer's diagram of their degree sequences. We show that the Laplacian spectra of threshold hypergraphs are $r$-integral, i.e., integral multiple of $r$, for some $r\in \mathbb{Q}$. We also construct another class of hypergraphs whose Laplacian spectra are $r$-integral.

math.CO↗
arXiv2023

On the spectral radius of some linear hypergraphs

Here we study the spectral radii of some linear hypergraphs, that is, the maximum moduli of the eigenvalues of their corresponding adjacency matrices. We determine the hypertrees having the largest to seventh-largest spectral radii. The hypertrees with the largest and the second-largest spectral radii among all those with a given diameter are identified here. Unicyclic hypergraphs with a fixed cycle length having the largest, the second-largest, and the third-largest spectral radii are also determined. Furthermore, we also find which bicyclic and tricyclic linear hypergraphs have the largest and the second-largest spectral radii.

math.CO↗
arXiv2023

A study of diffusion in network with multi-body interactions using Hypergraphs

Being cognizant of the abundance of multi-body interactions in various complex systems, here we investigate a possible way to incorporate multi-body interactions in dynamical networks. Adopting hypergraph as the underlying architecture aids our proposed dynamical network models to go beyond the traditional archetype of only pairwise interactions. We introduce some matrices associated with hypergraphs to incorporate multi-body frameworks in dynamic networks. We illustrate the fact that the approximation of multi-body interactions by pairwise binary interactions, i.e. considering graph as the underlying architecture of the corresponding dynamical network may lead to a wrong conclusion to the study. Here we use weighted hypergraphs to deal with the multi-body interactions of variable weights. We study the possibility of global and local synchronization in discrete and continuous-time dynamical networks. Some real-world numerical illustrations are included at the end to reinforce our theoretical results.

math.DS↗
arXiv2022

Symmetries of Hypergraphs and Some Invariant Subspaces of Matrices Associated with Hypergraphs

Here, the structural symmetries of a hypergraph are represented through equivalence relations on the vertex set of the hypergraph. A matrix associated with the hypergraph may not reflect a specific structural symmetry. In the context of a given symmetry within a hypergraph, we investigate a collection of matrices that encapsulate information about the symmetry. Our investigation reveals that certain structural symmetries in a hypergraph manifest observable effects on the eigenvalues and eigenvectors of designated matrices associated with the hypergraph. We identify specific matrices where the invariance is a consequence of symmetries present in the hypergraph. These invariant subspaces elucidate analogous behaviours observed in certain clusters of vertices during random walks and other dynamical processes on the hypergraph.

math.CO↗
arXiv2022

On some building blocks of hypergraphs

In this study, we explore the interrelation between hypergraph symmetries represented by equivalence relations on the vertex set and the spectra of operators associated with the hypergraph. We introduce the idea of equivalence relation compatible operators related to hypergraphs. Some eigenvalues and the corresponding eigenvectors can be computed directly from the equivalence classes of the equivalence relation. The other eigenvalues can be computed from a quotient operator obtained by identifying each equivalence class as an element. We provide an equivalence relation $\mathfrak{R}_s$ on the vertex set of a hypergraph such that the Adjacency, Laplacian, and signless Laplacian operators associated with that hypergraph become $\mathfrak{R}_s$-compatible. The $\mathfrak{R}_s$-equivalence classes are named as units. Using units, we find some more symmetric substructures of hypergraphs called twin units, regular sets, co-regular sets, and symmetric sets. We collectively classify them as building blocks of hypergraphs. We show that the presence of these building blocks leaves certain traces in the spectrum and the corresponding eigenspaces of the $\mathfrak{R}_s$-compatible operators associated with the hypergraph. We also show that, conversely, some specific footprints in the spectrum and the corresponding eigenvectors retrace the presence of some of these building blocks in the hypergraph. Besides the spectra of $\mathfrak{R}_s$-compatible operators, building blocks are also interrelated with hypergraph colouring, distances in hypergraphs, hypergraph automorphisms, and random walks on hypergraphs.

math.CO↗
arXiv2022

On Some General Operators of Hypergraphs

Here we introduce connectivity operators, namely, diffusion operators, general Laplacian operators, and general adjacency operators for hypergraphs. These operators are generalisations of some conventional notions of apparently different connectivity matrices associated with hypergraphs. In fact, we introduce here a unified framework for studying different variations of the connectivity operators associated with hypergraphs at the same time. Eigenvalues and corresponding eigenspaces of the general connectivity operators associated with some classes of hypergraphs are computed. Applications such as random walks on hypergraphs, dynamical networks, and disease transmission on hypergraphs are studied from the perspective of our newly introduced operators. We also derive spectral bounds for the weak connectivity number, degree of vertices, maximum cut, bipartition width, and isoperimetric constant of hypergraphs.

math.CO↗
arXiv2019

Joins of Hypergraphs and Their Spectra

Here, we represent a general hypergraph by a matrix and study its spectrum. We extend the definition of equitable partition and joining operation for hypergraphs, and use those to compute eigenvalues of different hypergraphs. We derive the characteristics polynomial of a complete $m$-uniform $m$-partite hypergraph $K^m_{n_1,n_2,\dots,n_m}$. Studying edge corona of hypergraphs we find the complete spectrum of $s$-loose cycles $C^m_{L(s;n)}$ for $m \geq 2s+1$ and the characteristics polynomial of a $s$-loose paths $P^{(m)}_{L(s;n)}$. Some of the eigenvalues of $P^{(m)}_{L(s;n)}$ are also derived. Moreover, using vertex corona, we show how to generate infinitely many pairs of non-isomorphic co-spectral hypergraphs.

math.CO↗
arXiv2019

On Minimum Order of Odd Regular Graphs Without Perfect Matching

In this article we have derived the minimum order of an odd regular graph such that the graph has no matching. We have observed that how it is different from the case of even regular graphs. We have checked the consistency of the derived result with Petersen's theorem.

math.CO↗
arXiv2019

On the spectrum of hypergraphs

Here we study the spectral properties of an underlying weighted graph of a non-uniform hypergraph by introducing different connectivity matrices, such as adjacency, Laplacian and normalized Laplacian matrices. We show that different structural properties of a hypergrpah, can be well studied using spectral properties of these matrices. Connectivity of a hypergraph is also investigated by the eigenvalues of these operators. Spectral radii of the same are bounded by the degrees of a hypergraph. The diameter of a hypergraph is also bounded by the eigenvalues of its connectivity matrices. We characterize different properties of a regular hypergraph characterized by the spectrum. Strong (vertex) chromatic number of a hypergraph is bounded by the eigenvalues. Cheeger constant on a hypergraph is defined and we show that it can be bounded by the smallest nontrivial eigenvalues of Laplacian matrix and normalized Laplacian matrix, respectively, of a connected hypergraph. We also show an approach to study random walk on a (non-uniform) hypergraph that can be performed by analyzing the spectrum of transition probability operator which is defined on that hypergraph. Ricci curvature on hypergraphs is introduced in two different ways. We show that if the Laplace operator, $Δ$, on a hypergraph satisfies a curvature-dimension type inequality $CD (\mathbf{m}, \mathbf{K})$ with $\mathbf{m}>1 $ and $\mathbf{K}>0 $ then any non-zero eigenvalue of $- Δ$ can be bounded below by $ \frac{ \mathbf{m} \mathbf{K}}{ \mathbf{m} -1 } $. Eigenvalues of a normalized Laplacian operator defined on a connected hypergraph can be bounded by the Ollivier's Ricci curvature of the hypergraph.

math.CO↗
arXiv2017

On the spectrum of directed uniform and non-uniform hypergraphs

Here, we suggest a method to represent general directed uniform and non-uniform hypergraphs by different connectivity tensors. We show many results on spectral properties of undirected hypergraphs also hold for general directed uniform hypergraphs. Our representation of a connectivity tensor will be very useful for the further development in spectral theory of directed hypergraphs. At the end, we have also introduced the concept of weak* irreducible hypermatrix to better explain connectivity of a directed hypergraph.

math.SP↗
arXiv2017

On the normalized spectrum of threshold graphs

In this article we investigate normalized adjacency eigenvalues (simply normalized eigenvalues) and normalized adjacency energy of connected threshold graphs. A threshold graph can always be represented as a unique binary string. Certain eigenvalues are obtained directly from its binary representation and the rest of the eigenvalues are evaluated from its normalized equitable partition matrix. Finally, we characterize threshold graphs with at most five distinct eigenvalues.

math.CO↗
arXiv2016

Network similarity and statistical analysis of earthquake seismic data

We study the structural similarity of earthquake networks constructed from seismic catalogs of different geographical regions. A hierarchical clustering of underlying undirected earthquake networks is shown using Jensen-Shannon divergence in graph spectra. The directed nature of links indicates that each earthquake network is strongly connected, which motivates us to study the directed version statistically. Our statistical analysis of each earthquake region identifies the hub regions. We calculate the conditional probability of the forthcoming occurrences of earthquakes in each region. The conditional probability of each event has been compared with their stationary distribution.

physics.geo-ph↗
arXiv2016

Spectra of general hypergraphs

Here, we show a method to reconstruct connectivity hypermatrices of a general hypergraph (without any self loop or multiple edge) using tensor. We also study the different spectral properties of these hypermatrices and find that these properties are similar for graphs and uniform hypergraphs. The representation of a connectivity hypermatrix that is proposed here can be very useful for the further development in spectral hypergraph theory.

math.SP↗
arXiv2015

On Extension of Regular Graphs

In this article, we discuss when one can extend an r-regular graph to an r + 1 regular by adding edges. Different conditions on the num- ber of vertices n and regularity r are developed. We derive an upper bound of r, depending on n, for which, every regular graph G(n, r) can be extended to an r + 1-regular graph with n vertices. Presence of induced complete bipartite subgraph and complete subgraph is dis- cussed, separately, for the extension of regularity.

math.CO↗
arXiv2015

Characteristics polynomial of normalized Laplacian for trees

Here, we find the characteristics polynomial of normalized Laplacian of a tree. The coefficients of this polynomial are expressed by the higher order general Randić indices for matching, whose values depend on the structure of the tree. We also find the expression of these indices for starlike tree and a double-starlike tree, $H_m(p,q)$. Moreover, we show that two cospectral $H_m(p,q)$ of the same diameter are isomorphic.

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