The Vertex-Harary Matrix

The reciprocal vertex-distance matrix, denoted by IK, is called the vertex-Harary matrix (PlavSid et al., 1993) in honor of the late Professor Frank Harary (New York, 1921-Las Cruces, New Mexico, 2004), the grandmaster of both graph theory and chemical graph theory. It can simply be obtained by replacing off-diagonal elements of the vertex-distance matrix D by their reciprocals (Balaban et al., 1992 Mihalic and Trinajstte, 1992 Plavsid et al., 1993 Ivanciuc et al., 1993 Todeschini and Consonni, 2000,2009 Lucid et al., 2012)  

A variant of the vertex-Harary matrix of a given graph is its squared version, denoted by ( D ), which is derived from the related vertex-distance matrix D by replacing its elements with the squares of their reciprocals (PlavSid et al, 1993 Consonni and Todeschini, 2012)  

The motivation for introduction of the Harary matrix was pragmatic. The aim was to make a distance index differing from the Wiener index in the contributions of the distant sites. Namely, they should have much smaller contributions than the nearer sites, since in many circumstances the distant sites influence each other much less than nearer sites. 

Soon after the introduction of the vertex-Harary index, the hyper-Harary index was proposed (Diudea, 1997). Then, the vertex-Harary index was extended to heterosystems (Ivanciuc et al., 1998), and finally, the modified vertex-Harary index was proposed and used (Lucid et al., 2002). The vertex-Harary index based on the matrix ( D ) was also successfully tested in several structure-property relationships (Mihalid and Trinajstic, 1992 Plavsic etal., 1993 Triniystid etal., 2001). The vertex-Harary matrix can also be used to derive a variant of the Balaban index (Balaban, 1989). This novel index we call the Harary-Balaban index (Nikolic et al., 2001a). 

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As an example, we give below the vertex-Harary matrix for the vertex-labeled graph Gi (see structure A in Figure 2.1) ... 

The edge-Harary matrix of a graph G, denoted by IK, is the vertex-Harary matrix of the corresponding line graph UG) ... 

We show below the edge-Harary matrix of Gj (see structure B in Figure 2.1), that is, the vertex-Harary matrix of L(G ) (see Figure 2.12). 

A set of three topological indices defined for a —> H-depleted molecular graph in terms of the distance matrix D, the squared Harary matrix the vector 8 of vertex degrees, and the vector p of ring degrees [Cao and Yuan, 2001 Yuan and Cao, 2003]. 

The reciprocal distance matrix, denoted as D (or Harary matrix, H, or vertex Harary matrix)... 

The vertex-adjacency matrix or binary matrix, denoted by A, of a vertex-labeled connected simple graph G with Vvertices is a square VxVmatrix, which is determined by the adjacencies of vertices in G (Harary, 1971) ... 

The permanent (also referred to as the positive determinant) of the vertex-adjacency matrix per A can be used to enumerate the number of Kekule structures K, or in the graph-theoretical terminology 1-factors (Harary, 1971 Cvetkovid et al., 1995) or dimers (Percus, 1969, 1971 Cvetkovid et al., 1995), of alternant structures (Mine, 1978 Cvetkovid et al., 1972, 1974a Kasum et al., 1981 Schultz et al., 1992 Cash, 1995 Torrens, 2002 Jiang et al., 2006) ... 

The standard distance matrix or the vertex-distance matrix (or the minimum path matrix) of a vertex-labeled connected graph G (Harary, 1971 Gutman and Polansky, 1986 Buckley and Harary, 1990 Trinajstid, 1992 Mihalic et al., 1992 Todeschini and Consonni, 2000,2009 Consonni and Todeschini, 2012), denoted by D, is a real symmetric VxV matrix whose elements are defined as... 

The diameter D of a graph G is the longest geodesic distance between any two vertices i and j in G, i.e., the largest [ D]y value in the vertex-distance matrix (Harary, 1971). 

The vertex-edge incidence matrix of a graph G, denoted by VE, is an unsymmetri-cal (and, in general, not a square) Vx E matrix, which is determined by the incidences of vertices and edges in G (Harary, 1971 Johnson and Johnson, 1972 Bondy and Murty, 1976 Rouvray, 1976 Chartrand, 1977 Trinajstid, 1992 Todeschini and Consonni, 2000, 2009) ... 

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... 

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