Detour matrix

Two matrices are particularly important, both of them based on the topological distance between vertices within a graph the distance matrix D(G) and the detour matrix A(G). The first contains as values the smallest number of steps from vertex i to vertex j, and the second contains as values the longest paths. For example, Equation (6.5) shows the D and A matrices of the DIOP ligand. 

Cluj delta matrix -> Cluj matrices Cluj-detour index Cluj matrices Cluj-detour matrix Cluj matrices CluJ-distance index - Cluj matrices Cluj-distance matrix -> Cluj matrices Cluj matrices (CJ)... 

A square symmetric Cluj-detour matrix CJA of dimension AxA, is obtained from the unsymmetric Cluj-detour matrix by the relation ... 

Derived from Cluj matrices, the reciprocal Cluj-distance matrix CJD and reciprocal Cluj-detour matrix CJA are the matrices whose elements are the reciprocal of the corresponding symmetric Cluj matrix elements [Diudea et ai, 1998 Diudea et ai, 1998] ... 

The detour matrix A of a graph 5 (or maximum path matrix) is a square symmetric Ax A matrix, A being the number of graph vertices, whose entry i-j is the length of the longest path from vertex v, to vertex Vy [Buckley and Harary, 1990 Ivanciuc... 

From the distribution of the element values in the i th row of the detour matrix, the maximum path degree sequence of the i th vertex is derived as a local vector-descriptor defined as ... 

The characteristic polynomial of the detour matrix is called the detour polynomial and is defined as ... 

A modified detour matrix was proposed by substituting diagonal zero elements with the length of the longest path from each vertex to itself (i.e. the size of the cycle containing the considered vertex). From this modified matrix, the same molecular descriptors defined above can be calculated [Rucker and Rucker, 1998]. 

Each entry i-j of the Ap matrix is calculated from the detour matrix A as the following ... 

From the detour matrix and the distance matrix, a combined matrix, called detour/ distance matrix A/D (or maximum/minimum path matrix), is defined as [Ivanciuc and Balaban, 1994b] ... 

It is a square unsymmetric Ax A matrix, where the upper triangle of the matrix contains the elements of the detour matrix (information about the longest paths) and the lower triangle contains the elements of the topological distance matrix (information about the shortest paths). 

The maximum/minimum path sum of the i th vertex, denoted by MmPVS, is a local vertex invariant defined as the sum of the lengths of the longest and shortest paths between vertex v, and any other vertex in the molecular graph. It is calculated as the sum of elements over the / th row and / th column in the A/D matrix, or, alternatively, as the sum of the - vertex distance degree o, calculated on the distance matrix D and the maximum path sum MPVS, of the / th vertex calculated on the detour matrix A ... 

For both detour and detour/distance matrices there can also be defined the reciprocal detour matrix A and the reciprocal detour/distance matrix A/D, as ... 

Opposite to the distance matrix is the detour matrix, where the entries correspond to the length of the longest path between the vertices. Other related topological matrices are the -> distance/distance matrix, the - detour/distance matrix and the - dis-tance/detour quotient matrix. 

The Harary index and hyper-Harary index [Diudea et al., 1998] are obtained from, respectively, the l -order sparse - reciprocal detour matrix ... 

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