Topological distances

Starting from the Nandy s representation of DNA sequences, the eigenvalues obtained from the geometric distance/topological distance quotient matrixand its increasing powers were used to perform similarity/diversity analysis of DNA sequences [Randic, 2000a],... 

The —> leading eigenvalues obtained from the geometric distance/topological distance quotient matrix and its higher order matrices were proposed to describe the sequence. In any case, the degeneracy of this approach still remains large. 

The quotient map matrix, denoted as Q, is defined in a similar way to the geometric distance/topological distance quotient matrix, whose entries are defined in terms of the ratio of distances between a pair of vertices measured through the space and along the bonds. The elements of the quotient matrix Q are formally defined as [Randic, 2001e Bajzer, Randic et al, 2003]... 

From the detour matrix and the distance matrix, a combined matrix, called detour-distance combined matrix AAD (or maximum-minimum path matrix or detour distance-topological distance combined matrix), is defined as [Ivanduc and Balaban, 1994b]... 

A variant of the distance/detour quotient matrix is the detour/distance quotient matrix (or detour distance/topological distance quotient matrix), denoted by A/D, vhose off-diagonal elements are the reciprocal of the distance/detour quotient matrix elements [Plavsic, Trinajstic et al., 1998] ... 

T D T/D Topographic distance/topological distance quotient matrix ... 

G D GAD Geometric distance-topological distance combined matrix... 

The reciprocal of this matrix, obtained by inverting its off-diagonal elements, is the resistance/distance quotient matrix S2/D (or resistance distance/topological distance quotient matrix), defined as... 

Local vertex invariant based on the adjacency matrix, atomic numbers, and vertex degrees Mean distance topological index for any graph Mean distance topological index for acyclic graphs Hosoya index... 

We consider here two kinds of distance/distance matrices the topographic distance/topological distance matrix and the corresponding reciprocal matrix. We denote the first matrix as D/ D and the second as D/ D. 

The topographic distance/topological distance matrix is an unsymmetric VxV matrix, in which the upper matrix-triangle represents the same part of the corresponding topographic matrix, and the lower matrix-triangle the same part of vertex-distance matrix (defined in Section 4.1) ... 

The topographic distance/topological distance matrix D/D of T2 is given by... 

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