The Edge-Vertex Incidence Matrix

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 vertex-edge incident matrix associated with G14 is given below  

A more general description of the vertex-edge incidence matrix can be given. A vertex-edge incidence matrix VE with rows and colunms labeled by members of two sets / (with the manbers denoted by i) and J (with the members denoted by j) of subgraphs of a graph G can be defined as 

The vertex-edge incidence matrix found some use in chanistry e.g., Balandin in 1940 employed this matrix, called the property matrix, in his study of the physical and chemical properties of molecnles (Balandin, 1940), though this work seems to have been largely overlooked (Randic and Trinajstid, 1994). More recently, a few information-theoretic indices have been based on this matrix (Bonchev and Trinajstid, 1977 Bonchev, 1983 Magnnson et al., 1983 Todeschini and Consonni, 2000, 2009). 

---

Derived from the H-depleted molecular graph, the vertex-edge incidence matrix, denoted by is a rectangular, usually unsymmetrical, matrix Ax B whose rows are the vertices (A) and columns the edges (B) of a graph [Bonchev and Trinajstic, 1977]. Its elements equal one if the vertex v is incident to the edge Cj, and zero otherwise... 

An interesting relationship between the vertex-edge incidence matrix, the edge-vertex incidence matrix and the Laplacian matrix L was found as... 

As an example, we give the vertex-edge incidence matrix of a branched tree T2 (Figure 3.1) ... 

The edge-vertex incidence matrix EV is an nnsymmetrical E x V matrix, which is the transpose of the vertex-edge incidence matrix VE. The EV matrix belonging to T2 (see Figure 3.1) is given below ... 

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

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

The Laplacian matrix is also related to the vertex-edge and edge-vertex incidence matrices. 

For computational purposes it is useful to distinguish the directed edges associated with the external inputs and outputs, which are usually specified, from those associated with the flows internal to the network. Denoting the net output from vertex i by and including henceforth only the internal edges in the incidence matrix M, we may restate Eq. (7) as... 

Derived from the -> H-depleted molecular graph, the incidence matrix [Bonchev and Trinajstic, 1977] is a rectangular matrix representation of a graph whose rows are the vertices (atoms, A) and columns are the edges (bonds, B), i.e. having a dimension AxB. Their elements are i,y = 1 if the edge ey is incident to the vertex V , otherwise... 

The simplest form to represent the chemical information contained in a molecular graph is by a -> matrix representation of a molecular structure. Examples are -+ adjacency matrix A, - edge adjacency matrix E, vertex - distance matrix D, -> edge distance matrix D, - incidence matrix I, - Wiener matrix W, -> Hosoya Z-matrix Z, - Cluj matrices CJ, - detour matrix A, - Szeged matrix SZ, -> distance/distance matrix DD, and - detour/distance matrix A/D. 

The simplest form to represent the chemical information contained in a molecular graph is by graph-theoretical matrices. Examples are adjacency matrix A, —> edge a acency matrix E, vertex distance matrix D, edge distance matrix D, incidence matrix I, —> Wienermatrix... 

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... 

---