The Cycle-Edge Incidence Matrix

The edge-cycle incidence matrix of a polycyclic graph G, denoted by EC, is an Ex Cn matrix (n being the size of the cycle), which is determined by the incidences of edges and cycles in G  

The corresponding edge-cycle incidence matrix is given below  

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Cycle matrices are particular incidence matrices, where each column represents a graph —> circuit. Two main cycle matrices are defined the vertex-cycle incidence matrix, denoted as whose rows are the A vertices and the edge-cycle incidence matrix, denoted as whose rows are... 

Based on total and mean information content, several topological information indices can be calculated both from the vertex-cycle matrix ( information indices on the vertex-cycle incidence matrix) and the edge-cycle matrix (—> information indices on the edge-cycle incidence matrix). 

The edge-cycle incidence matrix is a rectangular unsymmetrical matrix vhose rows represent the edges and columns the circuits of the graph. This matrix of dimension Bx, where is the graph cyclicity, is formally defined as... 

The cycle-edge incidence matrix, denoted as is the transpose of the edge-cycle incidence matrix. 

Information about atoms and bonds belonging to cycles is usually encoded by the vertex-cycle incidence matrix and the edge-cycle incidence matrix. 

Information indices on the edge-cycle incidence matrix 1 are listed below. 

This is derived from the edge-cycle incidence matrix and is based on the partition of matrix elements according to their equalities ... 

It is evident that the cycle-edge incidence matrix is the transpose of the edge-cycle incidence matrix. This matrix is presented because it is used in the counting formula for spanning trees of graphs (see Section 2.20). 

The cycle-edge incidence matrix Z is a c x e matrix (e and e being the numbers of cycles and edges, respectively), which is determined by the incidences of cycles and edges in G ... 

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