Cluj Matrices

The Cluj matrix is a square unsymmetrical V x V matrix, denoted by C, whose elements are defined as (Diudea, 1996a, 1997a, 1997b, 1997c, 1997d Diudea and Gutman, 1998 Diudea et al., 1998,2007)  

The unsymmetrical Cluj matrix C may be symmetrized to give the path-Cluj matrix PC and the edge-Cluj matrix C  

The Hadatnard matrix product of two matrices A and B of the same dimensions is defined as 

It should be noted that for acyclic graphs the pC and C matrices are identical to the PW and matrices, respectively. 

For cycle-containing graphs, the matrix is equal to the edge-Szeged matrix (Ivanciuc and Ivanciuc, 1999)  

---

Cluj delta matrix -> Cluj matrices Cluj-detour index Cluj matrices Cluj-detour matrix Cluj matrices CluJ-distance index - Cluj matrices Cluj-distance matrix -> Cluj matrices Cluj matrices (CJ)... 

By the same operation, the unsymmetric P order sparse Cluj-distance matrix CJDu is calculated from the unsymmetric CJDu matrix. Analogously, from the Cluj-detour matrices the corresponding order sparse matrices are defined as CJA and CJAu-For the above-defined sparse Cluj matrices, the following relationships hold for any graph ... 

From Cluj matrices molecular - Wiener-type indices are derived. 

Derived from Cluj matrices, the reciprocal Cluj-distance matrix CJD and reciprocal Cluj-detour matrix CJA are the matrices whose elements are the reciprocal of the corresponding symmetric Cluj matrix elements [Diudea et ai, 1998 Diudea et ai, 1998] ... 

EVtype descriptors -> van der Waals excluded volume method EVwhole descriptors - van der Waals excluded volume method excess electron polarizability -> electric polarization descriptors excess molar refractivity -> molar refractivity expanded distance Cluj matrices -> expanded distance matrices expanded distance indices -> expanded distance matrices expanded distance matrices... 

If M is one among the -> Cluj matrices, then expanded distance Cluj matrices are obtained. Next, if M is the Szeged matrix, then the expanded distance Szeged matrix is derived analogously, expanded distance Szeged property matrices [Minailiuc et al., 1998] and expanded distance walk matrices are derived from - Szeged property matrices and - walk matrices. 

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. 

Gutman, I. and Diudea, M.V. (1998a). Defining Cluj Matrices and Cluj Invariants. MATCH (Comm.Math.Comp.Chem.), 36. 

The Cluj matrices are defined for any graph and are, in general, unsymmetrical, except for some symmetric graphs. They can be symmetrized by the Hadamard matrix product with their transpose ... 

The Cluj matrices defined above, both symmetric and unsymmetrical, can be either path-Cluj matrices (U CJ and SC J ) when all the pairs of vertices of the graph are accounted for in the matrix calculation or edge-Cluj matrices (UCJ and SCJ ) if the only nonzero elements correspond to edges, that is, only pairs of adjacent vertices are accounted for. The edge-Cluj matrices can be obtained by the Hadamard product of the path-Cluj matrices and the adjacency matrix A SCJ, = SCJp A UCJ, = UCJp A... 

In trees, since there exists only one path joining any pair of vertices, Cluj-distance and Cluj-detour matrices coincide moreover, symmetric Cluj matrices, SCJD and SCJA, are equal to the —> Wiener matrix W (SCJDg= SCjAg = W and SCJDp = SCJA = W ). For cyclic graphs, Cluj-distance and Cluj-detour matrices are different, while Wiener matrices are not defined. 

Expanded distance Cluj matrices were also proposed [Diudea and Gutman, 1998] as a generalization of the expanded distance matrix and calculated by the Hadamard matrix product between the unsymmetrical path-Cluj matrices UCJp and the distance matrix D ... 

Cluj indices are Wiener-type indices calculated either on symmetric (SC J) or unsymmetrical (UCJ) Cluj matrices as [Diudea, 1997d Diudea, Parv et al, 1997b Diudea and Gutman, 1998 Katona and Diudea, 2003]... 

Moreover, from unsymmetrical Cluj matrices UCJ other invariants are the —> matrix sum indices MS calculated as ... 

Un symmetrica I and symmetric path-Cluj-distance (UCJDp, SCJDp) and path-Cluj-detour (UCjAp, SCjAp) matrices for ethylbenzene. Elements of the corresponding edge-Cluj matrices are highlighted in bold face. VS, and CS, are the row sum and column sums, respectively. ... 

Other Cluj indices can be derived from the Cluj polynomials. The Cluj polynomials are counting polYnondak defined on the basis of Cluj matrices as [Diudea, 2002a Diudea, Vizitiu et al., 2007]... 

The Cluj difference matrix, denoted as CJ, is obtained in the same way as the Cluj matrix CJ, but path contributions are calculated only on paths larger than one [Diudea, 1996b, 1997a Ivanciuc, Ivanciuc et al, 1997]. The Cluj difference matrix is calculated as difference between path-Cluj and edge-Cluj matrices ... 

These are molecular indices derived from —> reciprocal Cluj matrices. The Harary-type index is calculated from the reciprocal edge-Cluj-distance matrix CJD as... 

---