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Rahul Roy

Publications and source records attributed to Rahul Roy.

At least 19 recordsLinked to original sources
arXiv2026

A Vertex-Localized Positive Square-Energy Strengthening of Tur\'an's Theorem

Let $G$ be a graph of order $n$ with the adjacency eigenvalues $\lambda_1(G) \geq \dots \geq \lambda_n(G) $. Let $c(v)$ denote the maximum order of a clique containing vertex $v$. We prove the vertex-localized positive square-energy inequality \[ \sqrt{s_+(G)} \leq \sum_{v\in V}\left(1-\frac1{c(v)}\right), \] where \[ s_+(G)=\sum_{\lambda_i(G)>0}\lambda_i(G)^2. \] We also characterize equality. Apart from edgeless graphs, equality holds precisely for graphs obtained from a complete regular multipartite graph by adding an arbitrary number of isolated vertices. This settles a conjecture of Kannan, Kumar and Pragada.

math.CO↗
arXiv2026

Maximizing the algebraic connectivity of graphs of given order and size: a proof of a conjecture of Kolokolnikov

The algebraic connectivity of a graph $G$ is a well-studied graph invariant that is related to other properties of the graph such as connectivity and expansion. Given $n$ and $m$, $\alpha(n,m)$ is the maximum algebraic connectivity of a graph with $n$ vertices and $m$ edges. In 2015, Kolokolnikov conjectured that $\alpha(n,2n-4)=2$ for $n\geq 4$, and verified this claim computationally for $n \le 12$. In this paper, we prove Kolokolnikov's conjecture. We also show that $\alpha(n,3(n-3)) = 3$ is false in general. %Combined with the computational verification for $n \le 12$, this yields $\alpha(n,2n-4)=2$ for all admissible values of $n$.

math.CO↗
arXiv2026

Data-Driven Dynamic Assortment in Online Platforms: Learning about Two Sides

We study a dynamic assortment problem on a two-sided service platform with incomplete information and heterogeneous customers in a discrete-time setting. In each period, a customer arrives seeking service, and the platform chooses an assortment of sellers to display. The customer then proposes a transaction to at most one seller in the assortment according to a multinomial logit choice model. After a fixed number of periods, sellers review the proposals they have received and each chooses at most one customer according to another multinomial logit choice model, after which the cycle repeats. A key challenge is that the platform does not know the choice-model parameters of either customers or sellers in advance. To our knowledge, this is the first study of a dynamic assortment problem in which both sides' choice parameters are unknown. We develop a data-driven algorithm that learns these parameters while optimizing the platform's objective over time. We evaluate performance using regret, which measures revenue loss relative to a clairvoyant benchmark that knows all parameters and customer arrivals in advance. We show that the algorithm's worst-case regret grows polylogarithmically over time, and we derive a matching lower bound, establishing its rate optimality.

cs.LG↗
arXiv2026

Convergences for a Virus-like Evolving Population driven by Mutually-exciting Hawkes Processes

This paper presents a stochastic model motivated by the study of a virus-like evolving population with different mutation rates. This is a continuous time birth-death model: the birth processes are mutually-exciting Hawkes processes and the death process is also a Hawkes process. This structure for the births and the deaths does not allow, in general, to get the Markov property of the processes involved. But considering the couple given by the Hawkes processes and their intensities we are able to deduce the necessary and sufficient conditions for the Markov property of the couple. This property is the main tool to get the convergence results describing the behaviour of the population, and the existence of a phase transition at a critical fitness level.

math.PR↗
arXiv2026

Structural and extremal properties of $l_1$-Fiedler value

The algebraic connectivity $a(G)$, defined as the second smallest eigenvalue of the Laplacian matrix $L(G)$, admits a well-known variational characterization involving the minimization of a quadratic form subject to an $\ell_{2}$-norm constraint. In a recent work, Andrade and Dahl (2024) proposed an analogous formulation based on the $\ell_{1}$-norm, leading to the introduction of a new graph parameter $b(G)$, referred to as the $l_1$-Fiedler value. In this article, we undertake a detailed investigation of the structural and extremal properties of $b(G)$. We first derive a Nordhaus--Gaddum type inequality for $b(G)$. For trees, we determine both global maximizer and minimizers of $b(G)$, and present extremal constructions for trees with prescribed diameter, maximum degree, and number of pendant vertices. We further establish a connection between $b(G)$ and Laplacian matrices, and obtain a bound for $b(G)$ in terms of the edge connectivity, along with a complete characterization of the graphs attaining equality. We derive an explicit formula that describes the behaviour of $b(G)$ under the addition of pendant vertices. We also investigate the connection between $b(G)$ and the isoperimetric number.

math.CO↗
arXiv2025

Dynamical crossover between stretched- and compressed-exponential relaxation in a photoexcited crystal

Anomalous relaxation is one of the hallmarks of disordered systems. Following perturbation by an external source, many glassy, jammed and amorphous systems relax as a stretched or compressed exponential as a function of time. However, despite their ubiquity, the origins of and the connection between these phenomenological relaxation functions remains to be understood. Here, we observe a tunable crossover from stretched- to compressed-exponential relaxation by photoexciting single crystal Ca3Ru2O7 across a structural phase transition. We present a simple lattice model that shows how spatial inhomogeneity and local, strain-mediated interactions cooperate to produce the dynamical crossover. Our work reveals anomalous relaxation dynamics in an idealized single crystal material and establishes photoexcited solids as promising platforms for probing the mechanisms underlying anomalous relaxation.

cond-mat.str-el↗
arXiv2025

On the $l_\infty$-analog of Algebraic Connectivity

The algebraic connectivity of a graph, defined as the second smallest eigenvalue of its Laplacian matrix, admits a well-known variational characterization involving the $\ell_2$-norm. Motivated by the recent introduction of its $\ell_\infty$-analogue by Andrade and Dahl, we investigate the graph parameter $\gamma(G)$, obtained by replacing the $\ell_2$-norm with the $\ell_\infty$-norm in the corresponding optimization problem. We establish a simple and explicit combinatorial formula expressing $\gamma(G)$ as the ratio of the order of the graph to its maximum transmission, thereby providing a direct graph-theoretic interpretation of the parameter. As a consequence, we obtain a polynomial-time algorithm based on breadth-first search, significantly simplifying the previously known linear programming approach. We prove that $\gamma(G)$ characterizes graph connectivity and completely characterize all $\ell_\infty$-Fiedler vectors as the vectors \[ \left\{\pm\left(1-\gamma(G)d(u,\cdot)\right):u\in \mathcal{M}(G)\right\}, \] where $\mathcal{M}(G)$ denotes the set of vertices of maximum transmission. Furthermore, we derive bounds for $\gamma(G)$ in terms of several classical graph invariants, including the distance spectral radius, Wiener index, algebraic connectivity, and Cheeger constant. Finally, we establish a product formula for $\gamma(G)$ under Cartesian products of graphs, leading to explicit expressions for important graph families such as hypercubes, Hamming graphs, grid graphs, and torus graphs.

math.CO↗
arXiv2025

Phase transitions for a unidirectional elephant random walk with a power law memory II: Some sharper estimates

We continue our study of the unidirectional elephant random walk (uERW) initiated in {\it {Electron. Commun. Probab.}} ({\bf 29} 2024, article no. 78). In this paper we obtain definitive results when the memory exponent $\beta\in (-1, p/(1-p))$. In particular using a coupling argument we obtain the exact asymptotic rate of growth of $S_n$, the location of the uERW at time $n$, for the case $\beta\in (-1, 0] $. Also, for the case $\beta\in (0, p/(1-p))$ we show that $P(S_n \to \infty) \in (0,1)$ and conditional on $\{S_n \to \infty\}$ we obtain the exact asymptotic rate of growth of $S_n$. In addition we obtain the central limit theorem for $S_n$ when $\beta \in (-1, p/(1-p))$.

math.PR↗
arXiv2025

A novel approach to navigate the taxonomic hierarchy to address the Open-World Scenarios in Medicinal Plant Classification

In this article, we propose a novel approach for plant hierarchical taxonomy classification by posing the problem as an open class problem. It is observed that existing methods for medicinal plant classification often fail to perform hierarchical classification and accurately identifying unknown species, limiting their effectiveness in comprehensive plant taxonomy classification. Thus we address the problem of unknown species classification by assigning it best hierarchical labels. We propose a novel method, which integrates DenseNet121, Multi-Scale Self-Attention (MSSA) and cascaded classifiers for hierarchical classification. The approach systematically categorizes medicinal plants at multiple taxonomic levels, from phylum to species, ensuring detailed and precise classification. Using multi scale space attention, the model captures both local and global contextual information from the images, improving the distinction between similar species and the identification of new ones. It uses attention scores to focus on important features across multiple scales. The proposed method provides a solution for hierarchical classification, showcasing superior performance in identifying both known and unknown species. The model was tested on two state-of-art datasets with and without background artifacts and so that it can be deployed to tackle real word application. We used unknown species for testing our model. For unknown species the model achieved an average accuracy of 83.36%, 78.30%, 60.34% and 43.32% for predicting correct phylum, class, order and family respectively. Our proposed model size is almost four times less than the existing state of the art methods making it easily deploy able in real world application.

cs.AI↗
arXiv2024

The elephant random walk in the triangular array setting

Gut and Stadm\"{u}ller (2021, 2022) initiated the study of the elephant random walk with limited memory. Aguech and El Machkouri (2024) published a paper in which they discuss an extension of results by Gut and Stadtm\"{u}ller (2022) for an "increasing memory" version of the elephant random walk without stops. Here we present a formal definition of the process which has been hinted at Eq. (2.2) in Gut and Stadtm\"{u}ller (2022). This definition is based on the triangular array setting. We give a positive answer to the open problem in Gut and Stadtm\"{u}ller (2022) for the elephant random walk, possibly with stops. We also obtain the CLT for the supercritical case of this model.

math.PR↗
arXiv2024

Phase transitions for a unidirectional elephant random walk with a power law memory

For the standard elephant random walk, Laulin (2022) studied the case when the increment of the random walk is not uniformly distributed over the past history instead has a power law distribution. We study such a problem for the unidirectional elephant random walk introduced by Harbola, Kumar and Lindenberg (2014). Depending on the memory parameter $p$ and the power law exponent $\beta$, we obtain three distinct phases in one such phase the elephant travels only a finite distance almost surely, and the other two phases are distinguished by the speed at which the elephant travels.

math.PR↗
arXiv2023

Universal localization-delocalization transition in chiral-symmetric Floquet drives

Periodically driven systems often exhibit behavior distinct from static systems. In single-particle, static systems, any amount of disorder generically localizes all eigenstates in one dimension. In contrast, we show that in topologically nontrivial, single-particle Floquet loop drives with chiral symmetry in one dimension, a localization-delocalization transition occurs as the time $t$ is varied within the driving period ($0 \le t \le \td$). We find that the time-dependent localization length $\lloc(t)$ diverges with a universal exponent as $t$ approaches the midpoint of the drive: $\lloc(t) \sim (t - \td/2)^{-\nu}$ with $\nu=2$. We provide analytical and numerical evidence for the universality of this exponent within the AIII symmetry class.

cond-mat.dis-nn↗
arXiv2023

Localization renormalization and quantum Hall systems

The obstruction to constructing localized degrees of freedom is a signature of several interesting condensed matter phases. We introduce a localization renormalization procedure that harnesses this property, and apply our method to distinguish between topological and trivial phases in quantum Hall and Chern insulators. By iteratively removing a fraction of maximally-localized orthogonal basis states, we find that the localization length in the residual Hilbert space exhibits a power-law divergence as the fraction of remaining states approaches zero, with an exponent of $\nu=0.5$. In sharp contrast, the localization length converges to a system-size-independent constant in the trivial phase. We verify this scaling using a variety of algorithms to truncate the Hilbert space, and show that it corresponds to a statistically self-similar expansion of the real-space projector. This result accords with a renormalization group picture and motivates the use of localization renormalization as a versatile numerical diagnostic for quantum Hall systems.

cond-mat.mes-hall↗
arXiv2023

Stability of fractional Chern insulators with a non-Landau level continuum limit

The stability of fractional Chern insulators is widely believed to be predicted by the resemblance of their single-particle spectra to Landau levels. We investigate the scope of this geometric stability hypothesis by analyzing the stability of a set of fractional Chern insulators that explicitly do not have a Landau level continuum limit. By computing the many-body spectra of Laughlin states in a generalized Hofstadter model, we analyze the relationship between single-particle metrics, such as trace inequality saturation, and many-body metrics, such as the magnitude of the many-body and entanglement gaps. We show numerically that the geometric stability hypothesis holds for Chern bands that are not continuously connected to Landau levels, as well as conventional Chern bands, albeit often requiring larger system sizes to converge for these configurations.

cond-mat.str-el↗
arXiv2023

Dynamics of viable $f(R)$ dark energy models in the presence of Curvature-Matter interactions

In this study, we analyze the dynamics of the interaction between dark matter and curvature-driven dark energy in viable $f(R)$ gravity models using the framework of dynamical system analysis. We incorporate this interaction by introducing a source term in their respective continuity equations, given by $Q = \frac{\kappa^2 \alpha}{3H}\tilde{\rho}_{\rm m}\rho_{\rm curv}$, and examine two $f(R)$ gravity models that comply with local gravity constraints and cosmological viability criteria. Our findings reveal subtle modifications to fixed points and their stability criteria when compared to the conventional dynamical analysis of $f(R)$ gravity models without matter-curvature interactions as proposed and examined in prior literature. We determine the parameter limits associated with the stability criteria of critical points of the dynamical system for both the models. Additionally, the introduction of interaction reveals variations in the dynamical aspects of cosmic evolution, contingent on the range of values for the relevant model parameters. These results are consistent with the observed features of cosmic evolution within specific limits of the $f(R)$ models parameters and the coupling parameter $\alpha$. We also examine the evolutionary dynamics of the universe in the interacting scenario through cosmological and cosmographic parameters. Our analysis demonstrates that the interacting scenario comprehensively accounts for all the observed phases of the universe's evolution and also results in a stable late-time cosmic acceleration. Additionally, we've studied how the coupling parameter affects evolutionary dynamics, particularly its impact on the matter-to-curvature energy density ratio. Our findings suggest that the chosen form of interaction can also address the cosmic coincidence problem.

gr-qc↗
arXiv2023

Topological Triviality of Flat Hamiltonians

Landau levels play a key role in theoretical models of the quantum Hall effect. Each Landau level is degenerate, flat and topologically non-trivial. Motivated by Landau levels, we study tight-binding Hamiltonians whose energy levels are all flat. We demonstrate that in two dimensions, for such Hamiltonians, the flat bands must be topologically trivial. To that end, we show that the projector onto each flat band is necessarily strictly local. Our conclusions do not need the assumption of lattice translational invariance.

cond-mat.mes-hall↗
arXiv2023

Classification of unitary operators by local generatability

Periodically driven (Floquet) systems can exhibit possibilities beyond what can be obtained in equilibrium. Both in Floquet systems and in the related problems of discrete-time quantum walks and quantum cellular automata, a basic distinction arises among unitary time evolution operators: while all physical operators are local, not all are locally generated (i.e., generated by some local Hamiltonian). In this paper, we define the notion of equivalence up to a locally generated unitary in all Altland-Zirnbauer symmetry classes. We then classify noninteracting unitaries in all dimensions on this basis by showing that equivalence up to a locally generated unitary is identical to homotopy equivalence.

cond-mat.mes-hall↗