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.

This work presents a double asymmetric data hiding technique. The first asymmetry is created by hiding secret data in the complex region of the cover image and keep the smooth region unaffected. Then another asymmetry is developed by hiding a different number of secret bits in the various pixels of the complex region. The proposed technique uses the ant colony optimization (ACO) based technique for the classification of complex and smooth region pixels. Then, the variable least significant bits (VLSB) data hiding framework is used to hide secret bits in the complex region of the cover image. A distance-based substitution technique, namely increasing distance increasing bits substitution algorithm, is used to ensure the asymmetry in the number of hidden bits. The double asymmetric framework enhances the security of the hidden secret data and makes the retrieval of hidden information difficult for unauthorized users. The algorithm results in high-quality stego images, and the hidden information does not attract the human visual system (HVS).

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-theart denoising schemes. Furthermore, the proposed scheme can be applied to colour images as well.

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.

This article introduces a fast meshless algorithm for the numerical solution nonlinear partial differential equations (PDE) by Radial Basis Functions (RBFs) approximation connected with the Total Variation (TV)-based minimization functional and to show its application to image denoising containing multiplicative noise. These capabilities used within the proposed algorithm have not only the quality of image denoising, edge preservation but also the property of minimization of staircase effect which results in blocky effects in the images. It is worth mentioning that the recommended method can be easily employed for nonlinear problems due to the lack of dependence on a mesh or integration procedure. The numerical investigations and corresponding examples prove the effectiveness of the recommended algorithm regarding the robustness and visual improvement as well as peak-signal-to-noise ratio (PSNR), signal-to-noise ratio (SNR), and structural similarity index (SSIM) corresponded to the current conventional TV-based schemes.