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R. Balaji

Publications and source records attributed to R. Balaji.

9 recordsLinked to original sources
arXiv2025

Trees with extremal Laplacian eigenvalue multiplicity

Let $T$ be a tree. Suppose $\lambda$ is an eigenvalue of the Laplacian matrix of $T$ with multiplicity $m_{T}(\lambda)$. It is known that $m_{T}(\lambda) \leq p(T)-1$, where $p(T)$ is the number of pendant vertices of $T$. In this paper, we characterize all trees $T$ for which there exists an eigenvalue $\lambda$ such that $m_{T}(\lambda)=p(T)-1$. We show that such trees are precisely either paths, or there exists an integer $q$ such that if $\alpha$ and $\beta$ are two distinct pendant vertices, then the distance $d(\alpha,\beta)$ satisfies $d(\alpha, \beta) \equiv 2q ~{\rm{mod}}~(2q+1)$. As a consequence, we show that $1$ is an eigenvalue of $L_T$ with multiplicity $p(T)-1$ if and only if $d(\alpha,\beta) \equiv 2\,\mbox{mod}\, 3$ for all distinct pendant vertices $\alpha$ and $\beta$ of $T$.

math.CO↗
arXiv2022

Inverse formula for distance matrices of gear graphs

Distance matrices of some star like graphs are investigated in \cite{JAK}. These graphs are trees which are stars, wheel graphs, helm graphs and gear graphs. Except for gear graphs in the above list of star like graphs, there are precise formulas available in the literature to compute the inverse/Moore-Penrose inverse of their distance matrices. These formulas tell that if $D$ is the distance matrix of $G$, then $D^\dagger = -\frac{1}{2}L+uu'$, where $L$ is a Laplacian-like matrix which is positive semidefinite and all row sums equal to zero. The matrix $L$ and the vector $u$ depend only on the degree and number of vertices in $G$ and hence, can be written directly from $G$. The earliest formula obtained is for distance matrices of trees in Graham and Lovász \cite{GL}. In this paper, we obtain an elegant formula of this kind to compute the Moore-Penrose inverse of the distance matrix of a gear graph.

math.CO↗
arXiv2021

On resistance matrices of weighted balanced digraphs

Let $G$ be a connected graph with $V(G)=\{1,\dotsc,n\}$. Then the resistance distance between any two vertices $i$ and $j$ is given by $r_{ij}:=l_{ii}^† + l_{jj}^†-2 l_{ij}^†$, where $l_{ij}^†$ is the $(i,j)^{\rm th}$ entry of the Moore-Penrose inverse of the Laplacian matrix of $G$. For the resistance matrix $R:=[r_{ij}]$, there is an elegant formula to compute the inverse of $R$. This says that \[R^{-1}=-\frac{1}{2}L + \frac{1}{τ' R τ} ττ', \] where \[τ:=(τ_1,\dotsc,τ_n)'~~\mbox{and}~~ τ_{i}:=2- \sum_{\{j \in V(G):(i,j) \in E(G)\}} r_{ij}~~~i=1,\dotsc,n. \] A far reaching generalization of this result that gives an inverse formula for a generalized resistance matrix of a strongly connected and matrix weighted balanced directed graph is obtained in this paper. When the weights are scalars, it is shown that the generalized resistance is a non-negative real number. We also obtain a perturbation result involving resistance matrices of connected graphs and Laplacians of digraphs.

math.CO↗
arXiv2021

Generalized Euclidean distance matrices

Euclidean distance matrices (EDM) are symmetric nonnegative matrices with several interesting properties. In this article, we introduce a wider class of matrices called generalized Euclidean distance matrices (GDMs) that include EDMs. Each GDM is an entry-wise nonnegative matrix. A GDM is not symmetric unless it is an EDM. By some new techniques, we show that many significant results on Euclidean distance matrices can be extended to generalized Euclidean distance matrices. These contain results about eigenvalues, inverse, determinant, spectral radius, Moore-Penrose inverse and some majorization inequalities. We finally give an application by constructing infinitely divisible matrices using generalized Euclidean distance matrices.

math.FA↗
arXiv2020

On distance matrices of wheel graphs with odd number of vertices

Let $W_n$ denote the wheel graph having $n$-vertices. If $i$ and $j$ are any two vertices of $W_n$, define \[d_{ij}:= \begin{cases} 0 & \mbox{if}~i=j \\ 1 & \mbox{if}~i~ \mbox{and} ~j~ \mbox{are adjacent} \\ 2 & \mbox{else}. \end{cases}\] Let $D$ be the $n \times n$ matrix with $(i,j)^{\rm th}$ entry equal to $d_{ij}$. The matrix $D$ is called the distance matrix of $W_n$. Suppose $n \geq 5$ is an odd integer. In this paper, we deduce a formula to compute the Moore-Penrose inverse of $D$. More precisely, we obtain an $n\times n$ matrix $\widetilde{L}$ and a rank one matrix $ww'$ such that \[D^\dagger = -\frac{1}{2} \widetilde{L}+\frac{4}{n-1}ww'.\] Here, $\widetilde{L}$ is positive semidefinite, ${\rm rank}(\widetilde{L})=n-2$ and all row sums are equal to zero.

math.CO↗
arXiv2020

An inverse formula for the distance matrix of a wheel graph with even number of vertices

Let $n \geq 4$ be an even integer and $W_n$ be the wheel graph with $n$ vertices. The distance $d_{ij}$ between any two distinct vertices $i$ and $j$ of $W_n$ is the length of the shortest path connecting $i$ and $j$. Let $D$ be the $n \times n$ symmetric matrix with diagonal entries equal to zero and off-diagonal entries equal to $d_{ij}$. In this paper, we find a positive semidefinite matrix $\widetilde{L}$ such that ${\rm rank}(\widetilde{L})=n-1$, all row sums of $\widetilde{L}$ equal to zero and a rank one matrix $ww^T$ such that \[D^{-1}=-\frac{1}{2}\widetilde{L} + \frac{4}{n-1}ww^T. \] An interlacing property between the eigenvalues of $D$ and $\widetilde{L}$ is also proved.

math.CO↗
arXiv2017

Detecting Malignant TLS Servers Using Machine Learning Techniques

TLS uses X.509 certificates for server authentication. A X.509 certificate is a complex document and various innocent errors may occur while creating/ using it. Also, many certificates belong to malicious websites and should be rejected by the client and those web servers should not be visited. Usually, when a client finds a certificate that is doubtful using the traditional tests, it asks for human intervention. But, looking at certificates, most people can't differentiate between malicious and non-malicious websites. Thus, once traditional certificate validation has failed, instead of asking for human intervention, we use machine learning techniques to enable a web browser to decide whether the server to which the certificate belongs to is malignant or not ie, whether the website should be visited or not. Once a certificate has been accepted in the above phase, we observe that the website may still turn out to be malicious. So, in the second phase, we download a part of the website in a sandbox without decrypting it and observe the TLS encrypted traffic (encrypted malicious data captured in a sandbox cannot harm the system). As the traffic is encrypted after Handshake is completed, traditional pattern-matching techniques cannot be employed. Thus we use flow features of the traffic along with the features used in the above first phase. We couple these features with the unencrypted TLS header information obtained during TLS Handshake and use these in a machine learning classifier to identify whether the traffic is malicious or not.

cs.CR↗
arXiv2012

Simple expressions for the long walk distance

The walk distances in graphs are defined as the result of appropriate transformations of the $\sum_{k=0}^\infty(tA)^k$ proximity measures, where $A$ is the weighted adjacency matrix of a connected weighted graph and $t$ is a sufficiently small positive parameter. The walk distances are graph-geodetic, moreover, they converge to the shortest path distance and to the so-called long walk distance as the parameter $t$ approaches its limiting values. In this paper, simple expressions for the long walk distance are obtained. They involve the generalized inverse, minors, and inverses of submatrices of the symmetric irreducible singular M-matrix ${\cal L}=ρI-A,$ where $ρ$ is the Perron root of $A.$

math.CO↗