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A homeomorph of a graph G is a graph resulted by inserting vertices of degree 2 (Fig. 1.6). A graph G is labeled, G(Lb), when its points are distinguished (e.g. by their numbers) from those of the corresponding abstract graph. There are n! possibilities of numbering a graph of order n. Path Tree Star Cycle Fig. 1.1 A variety of graphs K1 K2 K3 ...
- Mircea Vasile Diudea
- 2018
Sunil Jayant Kulkarni. Graph Theory: Applications to Chemical Engineering and Chemistry Galore International Journal of Applied Sciences and Humanities (www.gkpublication.in) 18 Vol.1; Issue: 2; April-June 2017 Balaban discussed applications of graph theory in chemistry. [11] He dealt with definition, enumeration, and systematic
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Dec 16, 2013 · Mehmet Aziz Yirik Kumsal Ecem Colpan Saskia Schmidt Maria Sorokina C. Steinbeck. Chemistry, Computer Science. 2021. TLDR. This review first covers the history of chemical graph theory, then provides an overview of its various techniques and applications for CASE, and finally summarises modern tools usingchemical graph theory for CASE. Expand. 3.
ruo HOSOYA Ochanomizu. Univ. rsity Department of. nformation Sciences Bunkyo-ku, Tokyo 112, Japan1. Chemical graph theory1.1. PURPOSE AND PREMISESIn this book you are invited to the world of the application of the graph theory to chemistry, especially on the problem how the topology of a molecule determines its react.
- Haruo Hosoya
- 1994
- 1 Adjacency Matrix
- 2 Distance Matrix
- 3 Detour Matrix
- 4 Combinatorial Matrices
- 5 Wiener Matrices
- 6 Cluj Matrices
- 7 Distance-Extended Matrices
- 8 Walk Matrices
- 9 Reciprocal Matrices
- 10 Layer and Shell Matrices
Since early nineteenth century, a matrix A(G) has been associated to an organic molecule to show its atomic adjacency/connectivity (Sylvester 1874). This is a square table, of dimensions n × n, whose entries are A(G) characterizes a graph up to isomorphism. It allows the reconstruction of the graph. A(G) is symmetric vs. its main diagonal, so that ...
Distance matrix D(G) was introduced by Harary (1969). It is a square symmetric table, of dimensions n × n, whose entries are defined as The non-diagonal entries of this matrix are just the topological distance between i and j. Figure 1.11illustrates the distance matrix. The half sum of all entries in D(G) provides the well-known Wiener Wtopological...
In cycle-containing graphs, when the shortest path (i.e., geodesic) is replaced by the longest path between two vertices i and j, the Detour matrix Δ(G) can be constructed (Harary 1969; Diudea et al. 2002; Lukovits 1996; Amić and Trinajstić 1995) Figure 1.12shows a Detour matrix.
Two path-calculated matrices have been proposed (Diudea 1996; Diudea et al. 1998): the distance-path Dp (Fig. 1.13) and the detour-path Δp (Fig. 1.14), whose elements are combinatorially calculated from the classical matrices, distance D (or distance- edge ) and detour Δ (or detour-edge) In the above, np(i, j) is the number of internal paths (Klein...
Randić proposed the so-called Wiener matrix, W, and exploited it as a source of structural invariants, useful in QSPR/QSAR (Randić et al. 1993, 1994). In trees, the non-diagonal entries in such a matrix are defined as: where ni and nj denote the number of vertices lying on the two sides of the edge /path, e/p (having i and j as endpoints); when def...
Cluj matrices CJ(G) have been proposed by Diudea (Diudea 1997a; Diudea et al. 1997a, b; Janežič et al. 2007); they are defined on Cluj fragments CJi , j , p which collect vertices v lying closer to i than to j, the endpoints of a path p(i, j). These fragments represent the vertex proximities (see also Gutman 1994) of i vs. any vertex j, joined by t...
Diudea (1997a, b) has performed the Hadamard product on the unsymmetric Cluj matrix: [D • UCJ]i , j = [D]i , j • [UCJ]i , j to provide a new matrix, that shows in trees, the equalities: CS(D • UCJ) = CS(Dp ) and RS(D • UCJ) = RS(Wp ). This matrix (illustrated in Fig. 1.18) is a direct proof of the theorem of Klein et al. (1995): in trees , the sum ...
Diudea (1996, 1999) has proposed the walk matrix, W(M1M2M3), defined by the principle of a single endpoint characterization of a path where WM1i is the walk degree of the vertex i, of extent [M2]ij , weighted by the property collected in M1 and M3 (i.e., the ith row-sum of the matrix M1, raised to power [M2]ij and multiplied by the entries of M3); ...
In Chemical Graph Theory, the distance matrix accounts for through bond interactions of atoms in molecules. However, these interactions decrease as the distance between atoms increases. This reason led to the introduction, by the QSAR Group of Timisoara (Ciubotariu 1987; Ciubotariu et al. 2004) and next by the groups of Balaban (Ivanciuc et al. 199...
1.2.10.1 Layer Matrices
Layer matrices (Skorobogatov and Dobrynin 1988) have been proposed in connection with sequences of walks (Halberstam and Quintas 1982; Bonchev et al. 1989; Dobrynin 1993); they are built up on the layer partitions in a graph. Let G(v)k be the kth layer of vertices v lying at distance k, in the partition G(i) with ecci being the eccentricity of i. The entries in the layer matrix (of vertex property) LM (Diudea 1994, 2010; Diudea et al. 1994; Diudea and Ursu 2003) are defined as Layer matrix is...
1.2.10.2 Shell Matrices
Entries in the shell matrix ShM (of vertex pair property , Diudea and Ursu 2003) are defined as The shell matrix is a collection of the above defined entries: A shell matrix, ShM(G), will partition the entries of a square matrix according to the vertex-distance partitions in the graph. The zero column entries [ShM]i , 0 can be the diagonal entries of the info matrix. Tables 1.1, 1.2 and 1.3 exemplify the Shell matrix of Cluj matrices for G1.2.6 Let now consider the behavior of the edge-calcul...
1.2.10.3 Centrality Index
On the above layer/shell matrices, a centrality index (Diudea and Ursu 2003; Ursu and Diudea 2005) is calculated as This index allows the finding of the graph center (e.g. the vertex having the largest Ci value) and provides an ordering of graph vertices according to their centrality.
- Mircea Vasile Diudea
- 2018
Dec 11, 2013 · Molecular graphs, representing atoms and bonds, play a crucial role in the quantitative topological classification of irregularity [2,3] in chemistry, biology, and common networks..
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Mathematical Chemistry is defined in the IntroducThe Editors’ Introduction and Preface, already tion to the series as “. . . the field that concerns itself referred to, are most illuminating and are well with the novel and non-trivial application of worthy of study, and then the reader is led to mathematics to chemistry”.
