Two graphs having the same number of vertices connected in the same way are said to be isomorphic. Topological matrices representing such graphs will have the same eigen value spectrum. Hence isomorphic graphs are also known as co spectral or iso spectral graphs.Â  There are certain molecular graphs representing chemically different substances which are found to be co spectral graphs. Therefore the molecular descriptors calculated using topological matrices will not be able to distinguish between these molecules.Â  Under such circumstances, edge weighted topological matrices can be formulated so that, their eigen value spectrum will be unique. The edge weighted molecular graphs with bond length as the weight for the edges in adjacency and distance matrices distinguish between the molecules having similar molecular graphs. So different molecules having the same molecular graph will have the same topological indices, but the edge weighted adjacency and distance matrices have the ability to overcome this problem of degeneracy

RAMAN, SUNDARESWARAN

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            I S S N  2 3 2 1 - 8 0 7 X            V o l u m e  1 3  N u m b e r 7      J o u r n a l  o f  A d v a n c e s  i n  c h e m i s t r y  6322 | P a g e                                        F e b r u a r y  2 0 1 7                                         w w w . c i r w o r l d . c o m                                        REMOVAL  OF  DEGENERACY  IN THE  TOPOLOGICAL INDICES OF COSPECTRAL GRAPHS USING EDGE WEIGHTED MOLECULAR GRAPHS R. Sundareswarana, Mathavimanisekarb and S.Lalithac a Assistant Professor, Department of  Mathematics, SSN College of Engineering, Cheenai. b Associate Professor, Department of Physics, Fatima College, Madurai. c Associate Professor and Head (Retd.), Department of Physics, Fatima College, Madurai. E mail: sundareswaranr@ssn.edu.in Abstract                Two graphs having the same number of vertices connected in the same way are said to be isomorphic. Topological matrices representing such graphs will have the same eigen value spectrum. Hence isomorphic graphs are also known as co spectral or iso spectral graphs.  There are certain molecular graphs representing chemically different substances which are found to be co spectral graphs. Therefore the molecular descriptors calculated using topological matrices will not be able to distinguish between these molecules.  Under such circumstances, edge weighted topological matrices can be formulated so that, their eigen value spectrum will be unique. The edge weighted molecular graphs with bond length as the weight for the edges in adjacency and distance matrices distinguish between the molecules having similar molecular graphs. So different molecules having the same molecular graph will have the same topological indices, but the edge weighted adjacency and distance matrices have the ability to overcome this problem of degeneracy. Keywords: Co spectral graphs, topological indices, edge weighted topological matrices.   Introduction Graph theory is a branch of Mathematics that analyses any system on the basis of a fundamental quality known as connectivity between various objects or topology of the system. Graphs can be used to model many types of relations and processes in physical, biological, social and information systems. A chemical graph is a mathematical structure which provides a pictorial representation of  a molecule, taking into account the internal connectivity of atoms in the molecule through bonds. The molecular connectivity determines a number of properties of the chemical systems. Therefore   many   predictions can be made regarding the physico-chemical properties  of the molecules using chemical graph theory. Nenad Trinajstic [1] has shown that many quantum mechanical results can be derived or rederived using spectral properties of associated graphs. Saraswathi Vishveshwara, K.V.Brinda and N.Kannan [2] have used graph spectral properties to study the protein structure, protein dynamics and protein-protein interactions. Mathavi Manisekar and S.Lalitha have used the techniques of chemical graph theory to find the centre of amino acids [3]. G.Al Hagri, M.El. Marraki and M.Essalih [4] have shown the effectiveness of the degree distance of a graph as a successful structure descriptor and in the prediction of the boiling points and calculation of ultrasonic velocity in organic material.  Roberto Todeschini and Viviana Consonni [5] have discussed a number of molecular descriptors with appropriate examples known till 2009. Chemical graph theory finds applications in reaction networks also. Ga̒bor Szederke̒nyi [6] has illustrated that two different reaction networks with mass action kinetics may have the same differential equation description even if the source complexes are not the same in the reaction system. Any chemical / molecular graph having N vertices can be represented by an NxN symmetric matrix, known as topological matrix. . There are different kinds of topological matrices like adjacency matrix, distance matrix, reciprocal distance matrix, Laplacian matrix and so on [7]. The set of eigen values of such a matrix is known as the spectrum of the graph and the sum of the absolute eigen values is termed as the energy of the graph. The adjacency matrix [8] The most simple and effective matrix of a molecular graph is the Adjacency matrix (A). The adjacency matrix of a molecular graph having N vertices is a square N x N symmetric matrix which is defined as [A]ij=  1 if i≠j and eij∈ E(G) 0 if i=j or eij ∈ E(G)              (1) where eij is the edge between i and j and E (G) is the set of edges of the graph G. The Distance Matrix[8] The distance matrix (D)  of a molecular graph having N vertices is a real symmetric N x N square matrix, whose elements [D]ij represent the minimum distance or minimum length of the path between the vertices i and j in terms of  the number of edges between them.   min (l (pij))   if   i ≠ j  0 if i = j                   (2)  [D]ij  =             I S S N  2 3 2 1 - 8 0 7 X            V o l u m e  1 3  N u m b e r 7      J o u r n a l  o f  A d v a n c e s  i n  c h e m i s t r y  6323 | P a g e                                        F e b r u a r y  2 0 1 7                                         w w w . c i r w o r l d . c o m                                       where pij is the number of edges separating the vertices i and j.  Vertex and edge weighted topological matrix A vertex and edge weighted graph  [1] is a graph having one or more vertices or edges which can be distinguished from the other vertices and edges. A number (the weight) may be assigned to each one of such vertices and edges in a weighted graph. Such weights for a vertex might represent for example the atomic weights, number of free electrons, etc., and for the edge, the bond lengths, number of bonds, the length of a route, the capacity of a line, the energy required to move between locations along a route, etc., depending on the problem in hand.  The corresponding topological matrices are known as vertex or edge weighted matrices. Co spectral graphs    Two graphs having the same number of vertices connected in the same way are said to be isomorphic. Topological matrices representing such graphs will have the same eigen value spectrum. Hence isomorphic graphs are also known as co spectral or iso spectral graphs [1].  There are certain molecular graphs representing chemically different substances which are found to be co spectral graphs. Therefore the molecular descriptors calculated using topological matrices will not be able to distinguish between these molecules.  Under such circumstances, edge weighted topological matrices can be formulated so that, their eigen value spectrum will be unique. To illustrate this fact, a set of co spectral graphs representing the molecules Methane (CH4), Carbon tetra fluoride (CF4), Carbon tetra chloride (CCl4), Carbon tetra bromide (CBr4) and Carbon tetra iodide (CI4) are considered. Fig.1 is the molecular graph of all the above  mentioned molecules.                                                                   1                                                                3              2            4                                                         5                                                                                    Fig.1     The atoms are labelled as 1,2,3,4 and 5. Atom 2 is carbon atom. Atoms 1,3,4, and 5 represent  H, F, Cl, Br and I atoms in the respective molecules.      The adjacency matrix  of this graph is A =     0 1 0 0 01 0 1 1 10 1 0 0 00 1 0 0 00 1 0 0 0    The eigen values of this matrix are  -2, 0 ,0,0, 2.  The graph energy , E = 4, which is the same for all the five molecules, CH4, CCl4, CF4, CBr4 and CI4.  This degeneracy can be lifted if the edge weighted adjacency matrices are considered. Here, the actual bond length is substituted in the place of ‘1’ in the adjacency matrix. The length of C – H bond is 1.14 A⁰. The edge weighted adjacency matrix A(EW) , thus formulated for CH4 is given below.  A (EW)  =    0 1.14 0 0 01.14 0 1.14 1.14 1.140 1.14 0 0 00 1.14 0 0 00 1.14 0 0 0      Now the graph energy is 4.56. Table 1 gives the edge weighted adjacency matix energies of  CH4, CCl4, CF4, CBr4 and CI4 . These values are unique for each molecule.  Table 1: S.No. Molecule BondLength (A⁰) [9] Molecular weight Edge weighted Adjacency matrix energy 1. CH4 1.14 16.04                4.56 2. CF4 1.35 88.00                5.40      3. CCl4 1.77 153.82                7.08  4. CBr4 1.94 331.63                7.76  5. CI4 2.13 519.60                8.52             I S S N  2 3 2 1 - 8 0 7 X            V o l u m e  1 3  N u m b e r 7      J o u r n a l  o f  A d v a n c e s  i n  c h e m i s t r y  6324 | P a g e                                        F e b r u a r y  2 0 1 7                                         w w w . c i r w o r l d . c o m                                        A graph is drawn between the molecular weights (x) and the corresponding energies(y) of edge weighted adjacency matrices. Fig.2  shows the linear relationship between the molecular weight and graph energy.        Fig.2.  Edge weighted Adjacency Matrix Energy Vs Molecular weight                        for  CH4, CF4, CCl4,CBr4 and CI4 molecules  The edge weighted adjacency matrix energies for the molecules Fluoromethane (CH3F), Chloromethane (CH3Cl), Bromomethane (CH3Br) and Methyl iodide (CH3I) are evaluated and given in Table 2.  Table 2:         A graph is drawn between the molecular weights (x)  and the corresponding energies (y) of edge weighted adjacency matrices of chemical graphs of  the molecules  CH4, CH3F, CH3Cl, CH3Br and CH3I. Fig.3 shows the linear relationship between the molecular weight and graph energy.       Fig.3 Edge weighted Adjacency Matrix Energy Vs Molecular weight                                 for CH4, CH3F, CH3Cl,  CH3Br and CH3I molecules  Similarly in formulating the edge weighted distance matrix, the appropriate bond lengths are added to get the distance between the vertices. Thus the degeneracy in Wiener index can be lifted. The distance matrix of CH4 is D =    0 1 2 2 21 0 1 1 12 1 0 2 22 1 2 0 22 1 2 2 0    The energy of this matrix is 13.21. The Wiener number W, and Wiener index W are calculated using this matrix using the following definitions [5].                                                      W = ½   𝑑𝑁𝑗=1𝑁𝑖=1 ij                                                                                                   (1) S. No. Molecule Molecular weight Edge weighted  Adj, matrix energy 1. CH4 16.04 4.560 2. CH3F 34.03 4.784 3. C H3Cl 50.49 5.303  4. C H3Br 94.94 5.536 5. C H3I 141.94 5.808 468100 200 400 600Edge Wted.Adj.Matrix energyM.Wt (amu)4560 50 100 150Edge wted. Adj.Matrix enregyM.Wt(amu)            I S S N  2 3 2 1 - 8 0 7 X            V o l u m e  1 3  N u m b e r 7      J o u r n a l  o f  A d v a n c e s  i n  c h e m i s t r y  6325 | P a g e                                        F e b r u a r y  2 0 1 7                                         w w w . c i r w o r l d . c o m                                                                                                  W    = 2W / N (N-1)                                                                 (2)       Where N is the number of vertices and   dij  is the distance between the vertices in terms of number of edges.  The values are : Wiener number = 16 and Wiener Index, W.I. = 1.6. The edge weighted distance matrix of CH4 is DEW  =    0 1.14 2.28 2.28 2.281.14 0 1.14 1.14 1.142.28 1.14 0 2.28 2.282.28 1.14 2.28 0 2.282.28 1.14 2.28 2.28 0     The energy of this matrix is 15.06.  Table 3 gives the energies of the edge weighted distance matrix of CH4, CF4, CCl4, CBr4 and CI4. Table 3:          Fig: 4  Edge weighted Wiener index Vs  Molecular weight for CH4, CF4, CCl4, A graph is drawn between the molecular weight and the Wiener index calculated using the edge weighted distance matrices. Fig. 4 shows this  relationship.                                             CBr4 and CI4 molecules                Table 4 gives energies of the edge weighted distance matrices of the molecules CH3F, CH3Cl, CH3Br, CH3I,  and the  values of Wiener number and Wiener index for the same, which are proportional to the corresponding molecular weights. Table 4: S.No. Molecule Molecular weight (amu) Edge weighted Distance Matrix energy Edge weighted W.No. W.I. 1. CH4 16.04 15.06 18.24 1.82 2. CF4 88.00 17.00 21.40 2.14 3. CCl4 153.80 23.38  28.32 2.83 4. CBr4 331.60 25.62 31.04 3.10 5. CI4 519.60 28.14 34.08 3.41 S.No. Molecule Molecular weight  (amu) Edge weighted Dist. Matrix energy Edge weighted W.No. W.I. 1. CH4 16.04 15.060 18.24 1.824 2. CH3F 34.03 15.764 19.08 1.908 1.52.53.54.50 200 400 600Edge wted. WIM.Wt(amu)            I S S N  2 3 2 1 - 8 0 7 X            V o l u m e  1 3  N u m b e r 7      J o u r n a l  o f  A d v a n c e s  i n  c h e m i s t r y  6326 | P a g e                                        F e b r u a r y  2 0 1 7                                         w w w . c i r w o r l d . c o m                                            From Fig.5  it is seen that  the molecular weights and Edge weighted Wiener index of CH4, CH3F, CH3Cl,  CH3Br and CH3I molecules have a linear relationship with each other.           Fig. 5  Molecular weight Vs Edge weighted Wiener index                          for CH4, CH3F,  CH3Cl,  CH3Br and CH3I molecules Conclusion  Thus it has been shown that the edge weighted graphs with bond length as the weight for the edges in adjacency and distance matrices distinguish between the molecules having similar molecular graphs. References 1.  Nenad Trinajstić, Chemical Graph Theory, Volume I, CRC Press, Inc.1983. 2. Saraswathi Visveshwara, K.V.Brinda and N.Kannan, Protein Structure: Insights from Graph theory, Journal of Theoretical and Computational Chemistry, Vol.1, N0.1 (2002) 1-25. 3. Mathavi Manisekar and S.Lalitha, Graph Spectral method for finding the centre of amino acids, Journal of Basic and Applied Research International, 8(3), (2015) 184-189. 4. G. Al Hagri, M.El Marraki, M. Isalia,The degree distance of Certain Particular Graphs, Applied Mathematical Sciences, Vol.6, No. 18, (2012) 857-867. 5. Roberto Todeschini and Viviana Consonni, Molecular Descriptors for  Chemoinformatics, Vol I & II, Wiley-VGH, Second Edition, ISBN 978-3-527-31852-0. 6. Gábor Szederkényi, Comment on “identifiability of chemical reaction networks”  by G.Craciun and C. Pantea, J Math Chem 45, (2009) 1172–1174. 7. A. T. Balaban, Chemical Applications of graph theory, Academic Press, 1976. 8. Qian-Nan Hu, Yi-Zeng Liang and Kai-Tai Fang, The Matrix Expression, Topological Index and Atomic Attribute of Molecular Topological Structure,  Journal of Data Science 1(2003)  361-389.  9. CRC Handbook of Chemistry and Physics, 72nd Edition, 1991.     3. CH3Cl 50.49 17.224 20.76 2.076 4. CH3Br 94.94 17.830 21.44 2.144 5. CH3I 141.94 18.510 22.20 2.220 1.7522.252.50 50 100 150Edge wted. W.I.M.Wt(amu)

REMOVAL  OF  DEGENERACY  IN THE  TOPOLOGICAL INDICES OF COSPECTRAL GRAPHS USING EDGE WEIGHTED MOLECULAR GRAPHS

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REMOVAL OF DEGENERACY IN THE TOPOLOGICAL INDICES OF COSPECTRAL GRAPHS USING EDGE WEIGHTED MOLECULAR GRAPHS

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