arXiv subjects

Kamal Lochan Patra

Publications and source records attributed to Kamal Lochan Patra.

23 records · Page 2Linked to original sources

arXiv2019

Laplacian eigenvalues of the zero divisor graph of the ring $\mathbb{Z}_{n}$

We study the Laplacian eigenvalues of the zero divisor graph $Γ\left(\mathbb{Z}_{n}\right)$ of the ring $\mathbb{Z}_{n}$ and prove that $Γ\left(\mathbb{Z}_{p^t}\right)$ is Laplacian integral for every prime $p$ and positive integer $t\geq 2$. We also prove that the Laplacian spectral radius and the algebraic connectivity of $Γ\left(\mathbb{Z}_{n}\right)$ for most of the values of $n$ are, respectively, the largest and the second smallest eigenvalues of the vertex weighted Laplacian matrix of a graph which is defined on the set of proper divisors of $n$. The values of $n$ for which algebraic connectivity and vertex connectivity of $Γ\left(\mathbb{Z}_{n}\right)$ coincide are also characterized.

math.SP↗

arXiv2018

Vertex connectivity of the power graph of a finite cyclic group II

The power graph $\mathcal{P}(G)$ of a given finite group $G$ is the simple undirected graph whose vertices are the elements of $G$, in which two distinct vertices are adjacent if and only if one of them can be obtained as an integral power of the other. The vertex connectivity $κ(\mathcal{P}(G))$ of $\mathcal{P}(G)$ is the minimum number of vertices which need to be removed from $G$ so that the induced subgraph of $\mathcal{P}(G)$ on the remaining vertices is disconnected or has only one vertex. For a positive integer $n$, let $C_n$ be the cyclic group of order $n$. Suppose that the prime power decomposition of $n$ is given by $n =p_1^{n_1}p_2^{n_2}\cdots p_r^{n_r}$, where $r\geq 1$, $n_1,n_2,\ldots, n_r$ are positive integers and $p_1,p_2,\ldots,p_r$ are prime numbers with $p_1<p_2<\cdots <p_r$. The vertex connectivity $κ(\mathcal{P}(C_n))$ of $\mathcal{P}(C_n)$ is known for $r\leq 3$, see \cite{panda, cps}. In this paper, for $r\geq 4$, we give a new upper bound for $κ(\mathcal{P}(C_n))$ and determine $κ(\mathcal{P}(C_n))$ when $n_r\geq 2$. We also determine $κ(\mathcal{P}(C_n))$ when $n$ is a product of distinct prime numbers.

math.CO↗

arXiv2018

Vertex connectivity of the power graph of a finite cyclic group

Let $n=p_1^{n_1}p_2^{n_2}\ldots p_r^{n_r}$, where $r,n_1,\ldots, n_r$ are positive integers and $p_1,p_2,\ldots,p_r$ are distinct prime numbers with $p_1<p_2<\cdots <p_r$. For the cyclic group $C_n$ of order $n$, let $\mathcal{P}(C_n)$ be the power graph of $C_n$ and $κ(\mathcal{P}(C_n))$ be the vertex connectivity of $\mathcal{P}(C_n)$. It is known that $κ(\mathcal{P}(C_n))=p_1^{n_1} -1$ if $r=1$. For $r\geq 2$, we determine the exact value of $κ(\mathcal{P}(C_n))$ when $2ϕ(p_1\ldots p_{r-1})\geq p_1\ldots p_{r-1}$, and give an upper bound for $κ(\mathcal{P}(C_n))$ when $2ϕ(p_1\ldots p_{r-1}) < p_1\ldots p_{r-1}$, which is sharp for many values of $n$ but equality need not hold always.

math.CO↗

arXiv2016

Center, centroid and subtree core of trees

For $n\geq 5$ and $2\leq g\leq n-3,$ consider the tree $P_{n-g,g}$ on $n$ vertices which is obtained by adding $g$ pendant vertices to one degree $1$ vertex of the path $P_{n-g}$. We call the trees $P_{n-g,g}$ as path-star trees. We prove that over all trees on $n\geq 5$ vertices, the distance between center and subtree core and the distance between centroid and subtree core are maximized by some path-star trees. We also prove that the tree $P_{n-g_0,g_0}$ maximizes both the distances among all path-star trees on $n$ vertices, where $g_0$ is the smallest positive integer such that $2^{g_0}+g_0>n-1.$

math.CO↗

arXiv2010

Minimizing Laplacian spectral radius of unicyclic graphs with fixed girth

In this paper we consider the following problem: Over the class of all simple connected unicyclic graphs on $n$ vertices with girth $g$ ($n,g$ being fixed), which graph minimizes the Laplacian spectral radius? We prove that the graph $U_{n,g}$ (defined in Section 1) uniquely minimizes the Laplacian spectral radius for $n\geq 2g-1$ when $g$ is even and for $n\geq 3g-1$ when $g$ is odd.

math.CO↗