Search

For a connected graph $\mathcal {G}$ , the distance $d(u, v)$ between any two vertices $u$ , $v~\in ~\mathcal {V(G)}$ is defined as; $d(u,v)=\min _{\mathcal {P}_{u,v}}\{\text {length of}~\mathcal {P}_{uv}\}$ i.e the minimum length of path connecting any two vertices. Two vertices $u$ , $v$ of a graph $G$ are said to be resolved by a vertices $\mathcal {W}$ of a graph $\mathcal {G}$ if $d(u,w) \neq d(v,w)$ . An ordered set $\mathcal {W} = \{w_{1},w_{2},w_{3},\ldots,w_{k}\}\,\,\in \,\,V(G)$ is resolving set for a connected graph $G$ , if for any two vertices $u$ , $v$ there exists $w_{i} \in W$ such that $d(u,w_{i}) \neq d(v, w_{i})$ . The representation of vertices $v$ w.r.t $\mathcal {W}$ is represented by $r(v|\mathcal {W})$ which is $k$ -vector( $k$ -tuple) $(d(v,w_{1}),d(v,w_{2}),d(v,w_{3}),\ldots,d(v,w_{k}))$ . If the representation of all the vertices of graph $\mathcal {G}$ are different w.r.t $\mathcal {W}$ then $\mathcal {W}$ is the resolving set for $G$ [1]. A resolving set that contains minimum numbers of vertices is metric basis for $\mathcal {G}$ . The cardinality of smallest resolving set is the metric dimension of $\mathcal {G}$ , represented by $dim(\mathcal {G})$ [2], [3]. A family of connected graph $\mathcal {G}$ has unbounded metric dimension if $dim(\mathcal {G})$ depends on order of a graph. In this paper, we show that total graph of path power three and four have unbounded metric dimension. We also proved some results on edges of power of path graph.

We propose a new fractional-order (space and time) total variation regularized model for multiplicative noise removal in this research article. We use the regularly varying fuzzy membership degrees to characterize the likelihood of a pixel related to edges, texture regions, and flat regions to improve model efficiency. This approach is capable of maintaining edges, textures, and other image information while significantly reducing the blocky effect. We opt for the option of local actions. In order to efficiently find the minimizer of the prescribed energy function, the semi-implicit gradient descent approach is used (which derives the corresponding fractional-order Euler-Lagrange equations). The existence and uniqueness of a solution to the suggested variational model are proved. Experimental results show the efficiency of the suggested model in visual enhancement, preserving details and reducing the blocky effect while extracting noise as well as an increase in the PSNR (dB), SSIM, relative error, and less CPU time(s) comparing to other schemes.

The Digital Total Variation (DTV) scheme is a digitized energy regularization scheme used for image denoising. This technique takes advantage of being applied to arbitrarily located data points and also has the edge detective property. This article aims to introduce a novel meshless scheme using DTV filtering and Radial Basis Functions (RBFs) to solve the associated equation with the DTV model numerically which results in the image denoising to remove additive noise from image information. This meshless algorithm based on local collocation and Multiquadric Radial Basis Function. These appearances allow this algorithm not only to remove the additive noise from images but also to resolve the discontinuities sharply. It is also noticed that the proposed meshless scheme is simple, fast, computationally effective, requires simply post-processing, and can be easily implemented mathematically. Experimental results confirm that the peak signal-to-noise ratio, the structural similarity, signal-to-noise ratio, the visual effect, and the computational performance of this new meshless scheme are improved compared with state-of-the-art denoising schemes. Furthermore, the proposed scheme can be applied to colour images as well.

A family $\mathcal {G}$ of connected graphs is a family with unbounded metric dimension if $dim (\text{G})$ is not constant and depends on the order of graph. In this paper, we compute the metric dimension of the splitting graphs $S(P_{n})$ and $S(C_{n})$ of a path and cycle. We prove that the metric dimension of these graphs varies and depends on the number of vertices of the graph.