Let G be a connected Graph, and let D[G] denote the double Graph of G. In this paper, we first derive closed-form formulas for some distance based topological indices for D[G] in terms of G. Finally, these formulas are applied for several special kinds of Graphs, such as, the complete Graph, the path and the cycle.

The reciprocal degree distance (RDD), defined for a connected Graph G as vertex-degree- weighted sum of the reciprocal distances, that is, RDD (G) = S u, vÎV (G) dG (u)+dG (v)/dG (u, v). The reciprocal degree distance is a weight version of the Harary index, just as the degree distance is a weight version of the Wiener index. In this paper, we present exact formulae for the reciprocal degree distance of join, tensor product, strong product and wreath product of Graphs in terms of other Graph invariants including the degree distance, Harary index, the first Zagreb index and first Zagreb coindex. Finally, we apply some of our results to compute the reciprocal degree distance of fan Graph, wheel Graph, open fence and closed fence Graphs.

The distance matrix, distance eigenvalue, and distance energy of a connected Graph have been studied in detail in literature where as the study on distance seidel matrix associated with a connected Graph is in progress. The eigenvalues ∂ S 1 ≥ ∂ S 2 >. . . ∂S n of the distance seidel matrix D S (G) of a Graph G forms the distance seidel spectrum of G. We describe here the distance seidel spectrum of some types of subdivision related Graphs of a regular Graph in terms of its adjacency spectrum. We also derive analytic expressions for the distance seidel energy of S¯(Cp), the partial complement of the subdivision Graph of a cycle Cp and the distance seidel energy of S(Cp), the complement of the even cycle C2p

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distance-based clustering methods categorize samples by optimizing a global criterion, finding ellipsoid clusters with roughly equal sizes. In contrast, density-based clustering techniques form clusters with arbitrary shapes and sizes by optimizing a local criterion. Most of these methods have several hyper-parameters, and their performance is highly dependent on the hyper-parameter setup. Recently, a Gaussian Density distance (GDD) approach was proposed to optimize local criteria in terms of distance and density properties of samples. GDD can find clusters with different shapes and sizes without any free parameters. However, it may fail to discover the appropriate clusters due to the interfering of clustered samples in estimating the density and distance properties of remaining unclustered samples. Here, we introduce Adaptive GDD (AGDD), which eliminates the inappropriate effect of clustered samples by adaptively updating the parameters during clustering. It is stable and can identify clusters with various shapes, sizes, and densities without adding extra parameters. The distance metrics calculating the dissimilarity between samples can affect the clustering performance. The effect of different distance measurements is also analyzed on the method. The experimental results conducted on several well-known datasets show the effectiveness of the proposed AGDD method compared to the other well-known clustering methods.

The folded hypercube $FQ_n$ is the Cayley Graph Cay$(\mathbb{Z}_2^n,S)$, where $S=\{e_1,e_2,\dots,e_n\}\cup \{u=e_1+e_2+\dots+e_n\}$, and $e_i = (0,\dots, 0, 1, 0,$ $\dots, 0)$, with 1 at the $i$th position, $1\leq i \leq n$. In this paper, we show that the folded hypercube $FQ_n$ is a distance-transitive Graph. Then, we study some properties of this Graph. In particular, we show that if $n\geq 4$ is an even integer, then the folded hypercube $FQ_n$ is an $automorphic$ Graph, that is, $FQ_n$ is a distance-transitive primitive Graph which is not a complete or a line Graph.

Many Graphs are constructed from simpler ones by the use of operations on Graphs, and as a consequence, the properties of the resulting constructions are strongly related to the properties of their constituents. This paper is concerned with computing some distance-based Graph invariants for three constructions on Graphs namely double Graph, extended double cover, and strong double Graph.