arXiv subjects

Aniruddha Samanta

Publications and source records attributed to Aniruddha Samanta.

13 recordsLinked to original sources

arXiv2025

Spectral properties of distance Laplacian matrices of complex unit gain graphs

A complex unit gain graph ($ \mathbb{T} $-gain graph), $ \Phi=(G, \varphi) $ is a graph where the function $ \varphi $ assigns a unit complex number to each orientation of an edge of $ G $, and its inverse is assigned to the opposite orientation. In this article, we study several spectral properties of distance Laplacian matrices of $\mathbb{T}$-gain graphs. In particular, we establish a characterization for the balanced $ \mathbb{T}$-gain graph in terms of the nullity of gain distance Laplacian matrices. As an example, it is shown that two switching equivalent $ \mathbb{T} $-gain graphs need not imply that their distance Laplacian spectra are the same. However, we provide a necessary condition for which two switching equivalent $ \mathbb{T} $-gain graphs have the same distance Laplacian spectra. Furthermore, we present a lower bound for spectral radii of gain distance Laplacian matrices in terms of the winner index. In addition, we establish some upper bounds for spectral radii of gain distance Laplacian matrices and characterize the equalities.

math.CO↗

arXiv2025

On spectrally optimal duals for r-erasures of frames generated by graphs

In [9], authors studied spectrally optimal dual frames for 1-erasure and 2-erasures of frames generated by graph. In this paper, we study spectrally optimal dual frames for r-erasures. We show that the spectral radius of the error operator of unitary equivalent frames is same with respect to their respective canonical dual frames. We prove that if a frame is generated by a connected graph, then its canonical dual frame is the unique spectrally optimal dual frame for r-erasures. Further, we show that the canonical dual of frames generated by disconnected graphs are non-unique spectrally optimal dual frames for r-erasures.

math.FA↗

arXiv2025

Improved bound of graph energy in terms of vertex cover number

Let $ G $ be a simple graph with the vertex cover number $ \tau $. The energy $ \mathcal{E}(G) $ of $ G $ is the sum of the absolute values of all the adjacency eigenvalues of $ G $. In this article, we establish $ \mathcal{E}(G)\geq 2\tau $ for several classes of graphs. The result significantly improves the known result $ \mathcal{E}(G)\geq 2\tau-2c$ for many classes of graphs, where $ c $ is the number of odd cycles.

math.CO↗

arXiv2024

Unifying adjacency, Laplacian, and signless Laplacian theories

Let $G$ be a simple graph with associated diagonal matrix of vertex degrees $D(G)$, adjacency matrix $A(G)$, Laplacian matrix $L(G)$ and signless Laplacian matrix $Q(G)$. Recently, Nikiforov proposed the family of matrices $A_\alpha(G)$ defined for any real $\alpha\in [0,1]$ as $A_\alpha(G):=\alpha\,D(G)+(1-\alpha)\,A(G)$, and also mentioned that the matrices $A_\alpha(G)$ can underpin a unified theory of $A(G)$ and $Q(G)$. Inspired from the above definition, we introduce the $B_\alpha$-matrix of $G$, $B_\alpha(G):=\alpha A(G)+(1-\alpha)L(G)$ for $\alpha\in [0,1]$. Note that $ L(G)=B_0(G), D(G)=2B_{\frac{1}{2}}(G), Q(G)=3B_{\frac{2}{3}}(G), A(G)=B_1(G)$. In this article, we study several spectral properties of $ B_\alpha $-matrices to unify the theories of adjacency, Laplacian, and signless Laplacian matrices of graphs. In particular, we prove that each eigenvalue of $ B_\alpha(G) $ is continuous on $ \alpha $. Using this, we characterize positive semidefinite $ B_\alpha $-matrices in terms of $\alpha$. As a consequence, we provide an upper bound of the independence number of $ G $. Besides, we establish some bounds for the largest and the smallest eigenvalues of $B_\alpha(G)$. As a result, we obtain a bound for the chromatic number of $G$ and deduce several known results. In addition, we present a Sachs-type result for the characteristic polynomial of a $ B_\alpha $-matrix.

math.CO↗

arXiv2024

On spectrally optimal duals of frames generated by graphs

Recently, the concept of frames generated by graphs has been introduced in \cite{D}. In this paper, we study spectrally optimal dual frames of frames generated by graphs. We show that if the frame is generated by a connected graph, then its canonical dual frame is the unique spectrally optimal dual frame for $1$-erasure and $2$-erasures. Further, we show that the canonical dual frames of frames generated by disconnected graphs are non-unique spectrally optimal dual frames for $1$-erasure and $2$-erasures.

math.FA↗

arXiv2024

On walk-regular graphs and optimal duals of frames generated by graphs

Erasures are a common problem that arises while signals or data are being transmitted. A profound challenge in frame theory is to find the optimal dual frames ($OD$-frames) to minimize the reconstruction error if erasures occur. In this paper, we study the optimal duals of frames generated by graphs. First, we characterize walk-regular graphs. Then, it is shown that the diagonal entries of the Moore-Penrose inverse of the Laplacian matrix (or adjacency matrix) of a walk-regular graph are equal. Besides, we prove that connected graphs generate full spark frames. Using these results, we establish that the canonical dual frames are the unique $OD$-frames of a frame generated by a walk-regular graph. A sufficient condition under which the canonical dual frame is the unique $OD$-frame is known. Here, we establish that the condition is also necessary if the frame is generated by a connected graph.

math.CO↗

arXiv2023

Bounds and extremal graphs for the energy of complex unit gain graphs

A complex unit gain graph ($ \mathbb{T} $-gain graph), $ \Phi=(G, \varphi) $ is a graph where the gain function $ \varphi $ assigns a unit complex number to each orientation of an edge of $ G $ and its inverse is assigned to the opposite orientation. The associated adjacency matrix $ A(\Phi) $ is defined canonically. The energy $ \mathcal{E}(\Phi) $ of a $ \mathbb{T} $-gain graph $ \Phi $ is the sum of the absolute values of all eigenvalues of $ A(\Phi) $. For any connected triangle-free $ \mathbb{T} $-gain graph $ \Phi $ with the minimum vertex degree $ \delta$, we establish a lower bound $ \mathcal{E}(\Phi)\geq 2\delta$ and characterize the equality. Then, we present a relationship between the characteristic and the matching polynomial of $ \Phi $. Using this, we obtain an upper bound for the energy $ \mathcal{E}(\Phi)\leq 2\mu\sqrt{2\Delta_e+1} $ and characterize the classes of graphs for which the bound sharp, where $ \mu$ and $ \Delta_e$ are the matching number and the maximum edge degree of $ \Phi $, respectively. Further, for any unicyclic graph $ G $, we study the gains for which the gain energy $ \mathcal{E}(\Phi) $ attains the maximum/minimum among all $ \mathbb{T} $-gain graphs defined on $G$.

math.CO↗

arXiv2023

On the spectrum of complex unit gain graph

A $\mathbb{T}$-gain graph is a simple graph in which a unit complex number is assigned to each orientation of an edge, and its inverse is assigned to the opposite orientation. The associated adjacency matrix is defined canonically, and is called $\mathbb{T}$-gain adjacency matrix. Let $\mathbb{T}_{G} $ denote the collection of all $\mathbb{T}$-gain adjacency matrices on a graph $G$. In this article, we study the cospectrality of matrices in $\mathbb{T}_{G} $ and we establish equivalent conditions for a graph $G$ to be a tree in terms of the spectrum and the spectral radius of matrices in $\mathbb{T}_{G} $. We identify a class of connected graphs $\mathfrak{F^{'}}$ such that for each $G \in \mathfrak{F^{'}}$, the matrices in $\mathbb{T}_G$ have nonnegative real part up to diagonal unitary similarity. Then we establish bounds for the spectral radius of $\mathbb{T}$-gain adjacency matrices on $ G \in \mathfrak{F^{'}} $ in terms of their largest eigenvalues. Thereupon, we characterize $\mathbb{T}$-gain graphs for which the spectral radius of the associated $\mathbb{T}$-gain adjacency matrices equal to the largest vertex degree of the underlying graph. These bounds generalize results known for the spectral radius of Hermitian adjacency matrices of digraphs and provide an alternate proof of a result about the sharpness of the bound in terms of largest vertex degree established in [Krystal Guo, Bojan Mohar. Hermitian adjacency matrix of digraphs and mixed graphs. J. Graph Theory 85 (2017), no. 1, 217-248.].

math.CO↗

arXiv2021

On Weaving Generalized Frames and Generalized Riesz Bases

Weaving frames have potential applications in wireless sensor networks that require distributed processing of signal under different frames. In this paper, we study some new properties of weaving generalized frames (or $g$-frames) and weaving generalized orthonormal bases (or $g$-orthonormal bases). It is shown that a $g$-frame and its dual $g$-frame are woven. The inter-relation of optimal $g$-frame bounds and optimal universal $g$-frame bounds is studied. Further, we present a characterization of weaving $g$-frames. Illustrations are given to show the difference in properties of weaving generalized Riesz bases and weaving Riesz bases.

math.FA↗

arXiv2021

Gain distance matrices for complex unit gain graphs

A complex unit gain graph ($ \mathbb{T} $-gain graph), $ Φ=(G, φ) $ is a graph where the function $ φ$ assigns a unit complex number to each orientation of an edge of $ G $, and its inverse is assigned to the opposite orientation. %A complex unit gain graph($ \mathbb{T} $-gain graph) is a simple graph where each orientation of an edge is given a complex unit, and its inverse is assigned to the opposite orientation of the edge. In this article, we propose gain distance matrices for $ \mathbb{T} $-gain graphs. These notions generalize the corresponding known concepts of distance matrices and signed distance matrices. Shahul K. Hameed et al. introduced signed distance matrices and developed their properties. Motivated by their work, we establish several spectral properties, including some equivalences between balanced $ \mathbb{T} $-gain graphs and gain distance matrices. Furthermore, we introduce the notion of positively weighted $ \mathbb{T} $-gain graphs and study some of their properties. Using these properties, Acharya's and Stanić's spectral criteria for balance are deduced. Moreover, the notions of order independence and distance compatibility are studied. Besides, we obtain some characterizations for distance compatibility.

math.CO↗

arXiv2021

On the multiplicity of $Aα$-eigenvalues and the rank of complex unit gain graphs

Let $ Φ=(G, φ) $ be a connected complex unit gain graph ($ \mathbb{T} $-gain graph) on a simple graph $ G $ with $ n $ vertices and maximum vertex degree $ Δ$. The associated adjacency matrix and degree matrix are denoted by $ A(Φ) $ and $ D(Φ) $, respectively. Let $ m_α(Φ,λ) $ be the multiplicity of $ λ$ as an eigenvalue of $ A_α(Φ) :=αD(Φ)+(1-α)A(Φ)$, for $ α\in[0,1) $. In this article, we establish that $ m_α(Φ, λ)\leq \frac{(Δ-2)n+2}{Δ-1}$, and characterize the classes of graphs for which the equality hold. Furthermore, we establish a couple of bounds for the rank of $A(Φ)$ in terms of the maximum vertex degree and the number of vertices. One of the main results extends a result known for unweighted graphs and simplifies the proof in [15], and other results provide better bounds for $r(Φ)$ than the bounds known in [8].

math.CO↗

arXiv2020

Bounds for the energy of a complex unit gain graph

A $\mathbb{T}$-gain graph, $Φ= (G, φ)$, is a graph in which the function $φ$ assigns a unit complex number to each orientation of an edge, and its inverse is assigned to the opposite orientation. The associated adjacency matrix $ A(Φ) $ is defined canonically. The energy $ \mathcal{E}(Φ) $ of a $ \mathbb{T} $-gain graph $ Φ$ is the sum of the absolute values of all eigenvalues of $ A(Φ) $. We study the notion of energy of a vertex of a $ \mathbb{T} $-gain graph, and establish bounds for it. For any $ \mathbb{T} $-gain graph $ Φ$, we prove that $2τ(G)-2c(G) \leq \mathcal{E}(Φ) \leq 2τ(G)\sqrt{Δ(G)}$, where $ τ(G), c(G)$ and $ Δ(G)$ are the vertex cover number, the number of odd cycles and the largest vertex degree of $ G $, respectively. Furthermore, using the properties of vertex energy, we characterize the classes of $ \mathbb{T} $-gain graphs for which $ \mathcal{E}(Φ)=2τ(G)-2c(G) $ holds. Also, we characterize the classes of $ \mathbb{T} $-gain graphs for which $\mathcal{E}(Φ)= 2τ(G)\sqrt{Δ(G)} $ holds. This characterization solves a general version of an open problem. In addition, we establish bounds for the energy in terms of the spectral radius of the associated adjacency matrix.

math.CO↗

arXiv2019

On the adjacency matrix of a complex unit gain graph

A complex unit gain graph is a simple graph in which each orientation of an edge is given a complex number with modulus 1 and its inverse is assigned to the opposite orientation of the edge. In this article, first we establish bounds for the eigenvalues of the complex unit gain graphs. Then we study some of the properties of the adjacency matrix of complex unit gain graph in connection with the characteristic and the permanental polynomials. Then we establish spectral properties of the adjacency matrices of complex unit gain graphs. In particular, using Perron-Frobenius theory, we establish a characterization for bipartite graphs in terms of the set of eigenvalues of gain graph and the set of eigenvalues of the underlying graph. Also, we derive an equivalent condition on the gain so that the eigenvalues of the gain graph and the eigenvalues of the underlying graph are the same.

math.CO↗