[VP]ij= 
The vertex-path incidence matrix, denoted by VP, is variant of the vertex-edge incidence matrix, where all paths are taken into account instead of only paths of length one. The generalized definition of incidence matrices [177], already mentioned above in the case of the VE incidence matrix as an expression (38), in which the non-vanishing elements of the matrix represent non-zero intersection of two sets, is convenient to use to set up the VP matrices. Therefore, if V is the set of vertices {vi} and P the set of paths {pj} (a path being a sequence of edges such that each edge shares one vertex with the sequence-adjacent edges and shares no vertices with any other edge), than the VP incidence matrix is defined as:
[VP]ij= |
n(i,j) if {vi} and {pj} have non-zero intersections |
| 0 otherwise (44) |
where n(i,j) is the number of incidences between two sets {vi} and {pj}.
Below we give the vertex-path incidence matrix for T2.
p0 |
p1 |
p2 |
p3 |
p4 |
p5 |
||||
VP(T2)= |
v1 |
 |
1 |
1 |
2 |
2 |
1 |
1 |
 |
v2 |
1 |
3 |
2 |
1 |
1 |
0 |
|||
v3 |
1 |
3 |
3 |
1 |
0 |
0 |
|||
v4 |
1 |
2 |
3 |
2 |
0 |
0 |
|||
v5 |
1 |
2 |
1 |
2 |
2 |
0 |
|||
v6 |
1 |
1 |
1 |
1 |
2 |
2 |
|||
v7 |
1 |
1 |
2 |
3 |
1 |
0 |
|||
v8 |
1 |
1 |
2 |
2 |
1 |
1 |
The vertex-path incidence matrix is also termed the path-layer matrix [181-183], when the paths are organized with respect to length. The VP matrix is analogous to the so-called cardinality layer matrix [21] that has also been called the path-layer matrix [181], the distance-frequency matrix [184] and the path-sequence matrix [21]. Diudea and coworkers [185-187], Dobrynin [118] and others [e.g., 189,190] derived a number of layer matrices.
