The Detour-Complement Matrix

In parallel to the distance-complement matrix, the detour-complement matrix, denoted by DM, can be defined as 

The detour-complement matrix can be used to generate the detour-complement index. 

23 THE VERTEX-DISTANCE MATRIX AND THE DETOUR MATRIX OF COMPLETE GRAPHS AND COMPLETE BIPARTITE GRAPHS 

The vertex-distance matrix of a complete graph Ky is defined as 

Since E - is always eqnal to 1 in Ky graphs, the corresponding vertex-distance matrices have all elements identical and equal to unity. Consequently, the vertex-distance matrices of Ky graphs are identical to the vertex-adjacency matrices of these graphs. This is exemplified in the case of K.  

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The detour complement matrix AC for simple graphs is defined as [ Janezic, Milicevic et al., 2007]... 

The detour-complement matrix of the vertex-labeled graph Gj (structure A in Figure 2.1) is... 

Examples of complement matrices are the —> distance complement matrix, complementary distance matrix, —> reverse Wiener matrix, —> complement Barysz distance matrix, and detour complement matrix. 

Several quotient matrices are in use. Here we list six the vertex-distance/detour matrix D/DM (Randid, 1994b), the detour/vertex-distance matrix DMAD (Plavsid et al., 1998), the vertex-distance/resistance-distance matrix D/Q (Babid et al., 2002 Klein and Ivanciuc, 2002), the resistance-distance/vertex-distance matrix 12AD (Babid et al., 2002 Klein and Ivanciuc, 2002), the vertex-distance/vertex-distance-complement matrix DA D (Nikolid et al., 2001a), and the vertex-distance-complement/vertex-distance matrix " DAD (Nikolid et al, 2001a). These six quotient matrices for Gj (see structure A in Figure 2.1) are given below ... 

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