The Vertex-Path Incidence Matrix

The vertex-path incidence matrix is also termed the path-layer matrix (Skorobogatov and Dobrynin, 1988 Diudea, 1994,2003), when the paths are organized with respect to length. The VP matrix is analogous to the so-called cardinality layer matrix (Todeschini and Consonni, 2000, 2009), which has also been called the path-layer matrix (Skorobogatov and Dobrynin, 1988), the distance-frequency matrix (Diudea and F rv, 1988), and the path-sequence matrix (Todeschini and Consonni, 2000, 2009). Diudea and coworkers (Diudea et al., 1991, 1994 Diudea and Ursu, 2003), Dobrynin (1993), and others (e.g., Hu and Xu, 1996 Yang et al., 2002) derived a number of layer matrices. 

6 THE WEICHTED-HEXACON-KEKULE-STRUCTURE INCIDENCE MATRIX 

Randid (2004) introduced a novel description of Kekule structures of benzenoids by replacing their standard representation by what he called the algebraic representation. 

The incidence matrix that connects the numeric and geometric Kekule structures can be constructed using expression (3.1). It is called the weighted-hexagon-Kekule-structure incidence matrix and is denoted by HK. 

FIGURE 3.5 Numeric Kekul structures of chrysene with the weighted hexagons. 

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The vertex-path inddence matrix, denoted as is an extension of the vertex-edge incidence... 

The degree of vertex is equivalent to the number of edges. Equation (1) describes the vertex-to-vertex adjacency. It is an Ny X Ny symmetric matrix having zero diagonal elements. Equation (2), however, defines incidence matrix that outlines vertices and edges. Lastly, equation (3) defines the path matrix that stores information about all paths that emanate from the root. It is a -1) matrix excluding... 

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