The Detour-Path Matrix

The detour-path matrix, denoted by DM, can similarly be defined as the vertex-distance-path matrix that is, the matrix DM is a square symmetric L x L matrix whose off-diagonal elements i,j count all paths of any length that are included within the longest path between vertex i and vertex j (Diudea, 1996a). Each element i,j of the DM is computed from the corresponding detour matrix as follows  

The hyper-detour index can be obtained from the detour-path matrix. For acyclic graphs, the detour-path matrix is equal to the distance-path matrix, and consequently, the hyper-detour index for acyclic graphs is equal to the hyper-distance-path index obtained from the distance-path matrix. The hyper-detour index has been used in the structure property modeling of lower acyclic and cyclic saturated hydrocarbons with up to eight carbon atoms (Trinajstic et al., 2001). 

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The detour-path matrix Ap, analogously defined as the -> distance-path matrix Dp is a square symmetric matrix Ax A whose off-diagonal entry i-j is the count of all paths of any length m ( < m < A,) that are included within the longest path from vertex v, to vertex vy (A,y) [Diudea, 1996a]. The diagonal entries are zero. 

The hyper-detour index ww can be obtained applying the Wiener operator to the detour-path matrix, as ... 

For acyclic graphs the hyper-distance-path index Dp coincides with the - hyper-Wiener index WW derived from the - Wiener matrix and with the -> hyper-detour index derived from the detour-path matrix. Moreover, it was proposed as an extension of the hyper-Wiener index for any graph [Klein et al., 1995],... 

Each entry ir-j of the detour-path matrix is calculated from the detour matrix A as the following ... 

The detour-delta matrix, denoted as A, is another combinatorial matrix derived as the difference between the detour-path matrix Ap and the detour matrix A [Janezic, Milicevic... 

The detour-path matrix of Gj (see structure A in Figure 2.1) is given below ... 

For acyclic graphs, the hyper-detour index ww is equal to the index Dp obtained from the distance-path matrix Dp and to the WW obtained from the - Wiener matrix W. 

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. 

Detour-path matrix and detour-delta matrix for the H-depleted molecular graph of ethylbenzene tutu is the hyperdetour index and W/(Aa) is the Wiener-type index derived from the detour-delta matrix. 

The most common combinatorial matrices are derived from the distance matrix and detour matrix these are the distance-path matrix, detour-path matrix, delta matrix, and... 

The detour matrix (or the maximum path matrix) of a vertex-labeled connected graph G, denoted by DM, is a real symmetric VxVmatrix whose elements are defined as (Harary, 1971 Buckley and Harary, 1990 Amic and Trinajstid, 1995, Trinajstid et al., 1997a Nikolic et al., 1999 Todeschini and Consonni, 2000, 2009)... 

The detour-delta matrix enumerates the number of all longest paths larger than unity between vertices i and j in a graph. The DM matrix of Gj (see structure A in Figure 2.1) is given below ... 

The detour matrix (or the maximum path matrix MP) element [A],y is defined as the length of the longest path between the two vertices v and vy ... 

The detour-distance matrix A-D defined by Ivanciuc and Balaban (originally called the maximum/minimum path matrix MmP) collects in its upper triangle the elements of the detour matrix while the lower triangle elements are identical to those in the distance 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. 

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... 

This definition is exactly the opposite of the definition of the - distance matrix whose off-diagonal elements are the lengths of the shortest paths between the considered vertices. However, the distance and detour matrices coincide for acyclic graphs, there being only one path connecting any pair of vertices. 

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 ... 

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]. 

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 ... 

The distance/detour quotient matrix D/A is also derived from detour and distance matrices but it is a square symmetric matrix AxA whose off-diagonal entries are the ratios of the lengths of the shortest to the longest path between any pair of vertices [Randic, 1997c]. It is defined as ... 

Criterion IP = minimum —> vertex path eccentricity (i.e., the largest distance from the ith vertex in the detour matrix) ... 

Detour matrix A for the H-depleted molecular graph of ethylbenzene. is the atom detour eccentricity, MPVSj is the maximum path sum of the ith vertex, and w is the detour index. 

A combined molecular index, called detour-Wiener combined index (or MmPS topological index) and denoted as wA W, is defined as the sum of the lengths of the longest and shortest paths betv een any two vertices in the molecular graph and is calculated from the detour-distance combined matrix as... 

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