- Email: [email protected]
On relationships between vertex-degrees, path-numbers and graph valence-shells in trees
18 March 2002
Chemical Physics Letters 354 (2002) 417–422 www.elsevier.com/locate/cplett
On relationships between vertex-degrees, path-numbers and graph valence-shells in trees Istv an Lukovits a, Sonja Nikolic b, Nenad Trinajstic a
b,*
Chemical Research Center, Hungarian Academy of Sciences, P.O. Box 17, H-1525 Budapest, Hungary b The Rugjer Boskovi c Institute, P.O. Box 180, HR-10002 Zagreb, Croatia Received 21 January 2002
Abstract It was shown that generalized vertex-degrees, path-numbers and graph valence-shells are closely related invariants in (chemical) trees. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction This Letter is motivated by a recent work of Randic [1] in which he has proposed a set of new graph invariants for chemical trees (which appears to be closely related to Diudea’s layer matrices [2,3]), calling them graph valence-shells. He has also shown that there exists a relationship between valence-shells and path-numbers. Our aim is to show that Randic’s result is a special case of a more general theorem.
2. Preliminaries The most elementary constituents of a (molecular) graph are vertices, edges, vertex-degrees, walks and paths [4]. They are the basis of many
*
Corresponding author. Fax: +385-1-468-0245. E-mail address: [email protected] (N. Trinajstic).
graph-theoretical invariants referred to (somewhat imprecisely) as topological indices [5], which have found considerable use in structure–property–activity modeling [6–9]. Zagreb indices [10–12] and the vertex-connectivity indices [13,14], which are given only in terms of vertex-degrees – although in different algebraic forms – are good examples. Structural formulas of molecules are converted into graphs simply by replacing words ‘atoms’ with ‘vertices’ and ‘bonds’ with ‘edges’ [15]. Trees are used to represent acyclic molecular structures. The degree dðiÞ of a vertex i (the vertex-degree) of a graph G is the number of adjacent vertices. However, we can redefine the vertex-degree dðiÞ by introducing the notion of the first neighbor [16,17]. All vertices among which the distance is unity (i.e., the adjacent vertices) are first neighbors. Thus, we can call dðiÞ the first-order vertex-degree, denote it by d1 ðiÞ and define it as the number of the first neighbors. The concept of neighbors allows the generalization of the concept of (first-order) vertex-degree to higher-order vertex-degrees. The
0009-2614/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 2 ) 0 0 1 5 3 - 7
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I. Lukovits et al. / Chemical Physics Letters 354 (2002) 417–422
second-; third-; . . . ; kth-order vertex-degree, d2 ðiÞ; d3 ðiÞ; . . . dk ðiÞ, are the number of second; third; . . . ; kth neighbors, respectively. The hydrogen-depleted tree T representing the carbon skeleton of 2,3,3-trimethylpentane and the corresponding first-order and higher-order vertex-degrees are shown in Fig. 1. A path is a sequence of adjacent edges, which do not pass through the same vertex more than once [4]. The length of the path will be denoted by k. Path-number Pk denotes the number of paths of length k. Already almost fifty years ago Platt [18] advocated the use of path-numbers in structure– property modeling. However, this concept was already used by Wiener [19], who named the number of distances equal to three ‘polarity numbers’. However, the significance of Wiener’s and Platt’s
invention was overlooked until resurrection of chemical graph theory in the early seventies [5,20]. Recently several papers appeared in which quantitative structure–property relationships (QSPR) modeling was performed using path-numbers [21– 23]. In most cases QSPR models based on pathnumbers possessed better statistical parameters as compared to models based on other kinds of molecular descriptors. Therefore, path-numbers appear to be molecular descriptors that may play important role both in QSPR and quantitative structure–activity relationships (QSAR) modeling. This fact alone warrants the current interests in the properties of these numbers [1–3]. In Fig. 2 we give path-numbers for the 2,3,3-trimethylpentane tree shown in Fig. 1. Note that path-number P1 is equal to the number of edges E in a tree.
Fig. 1. Tree T representing the carbon skeleton of 2,3,3-trimethylpentane and the corresponding first- and higher-order vertex-degrees.
I. Lukovits et al. / Chemical Physics Letters 354 (2002) 417–422
419
Fig. 2. Path-numbers for the 2,3,3-trimethylpentane tree. Thick lines denote paths.
3. Vertex-degrees and path-numbers There is a simple relationship between vertexdegrees and path-numbers, which is obvious from Figs. 1 and 2: X Pk ¼ ð1=2Þ dk ðiÞ: ð1Þ i
This relationship is valid only for trees. Eq. (1) represents a generalization of the handshaking lemma, proved by Euler [24] in 1736: X 2E ¼ d1 ðiÞ: ð2Þ i
A consequence of the close relationship between vertex-degrees and path-numbers is that first- and higher-order vertex-degrees may replace pathnumbers in QSPR models for acyclic molecules with no effect on statistical parameters of the model as shown by Randic [1].
Both vertex-degrees and path-numbers can be straightforwardly obtained from the distance matrix [15]. A very efficient algorithm, developed by M€ uller et al. [25], is available for computing the distance matrix of any graph. Once we possess the distance matrix of a tree, we can easily obtain the total vertex-degree of the kth order, Dk , which is equal to the number of distances of the k-length. Path-numbers can be obtained in the similar way from the upper (or the lower) triangle of the distance matrix: a given path-number Pk is equal to the number of distances of length k in the triangle. The correctness of the calculated path-numbers can simply be checked by using Eq. (1). It should also be noted that the total vertex-degree of order k is given by expression: X Dk ¼ dk ðiÞ: ð3Þ i
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I. Lukovits et al. / Chemical Physics Letters 354 (2002) 417–422
Table 1 The distance matrix and the corresponding total vertex-degrees and the path-numbers of the 2,3,3-trimethylpentane tree shown in Fig. 1 0 1 2 3 4 2 3 3
1 0 1 2 3 1 2 2
2 1 0 1 2 2 1 1 D1 D2 D3 D4
3 2 1 0 1 3 2 2 ¼ 14 ¼ 20 ¼ 18 ¼4
4 3 2 1 0 4 3 3 P1 P2 P3 P4
2 1 2 3 4 0 3 3
3 2 1 2 3 3 0 2
3 2 1 2 3 3 2 0
¼7 ¼ 10 ¼9 ¼2
In Table 1 we give the distance matrix and the corresponding total vertex-degrees and pathnumbers of the 2,3,3-trimethylpentane tree shown in Fig. 1.
The calculation of Sk for given k can be illustrated by using as an example the 2,3,3-trimethylpentane tree shown in Fig. 1. For example, the value of the S3 can be obtained as follows: (i) the valence-shell of vertex 1 is 4 (sum of degrees of vertices 4, 7 and 8), (ii) the valence-shell of vertex 2 is 1 (degree of vertex 5), (iii) the valence-shell of vertex 3 is 0 (there is no vertex at distance equal to three from vertex 3), (iv) the valence-shell of vertex 4 is 2 (sum of degrees of vertices 1 and 6), (v) the valence-shell of vertex 5 is 5 (sum of degrees of vertices 2, 7 and 8), (vi) the valence-shell of vertex 6 is 4 (sum of degrees of vertices 4, 7 and 8), (vii) the valence-shell of vertex 7 is 3 (sum of degrees of vertices 1, 5 and 6), (viii) the valence-shell of vertex 8 is 3 (sum of degrees of vertices 1, 5 and 6). Therefore,
3.1. Generalization of Randi c’s result
S3 ¼ ð4 þ 1 þ 0 þ 2 þ 5 þ 4 þ 3 þ 3Þ=2 ¼ 11:
A valence-shell sk ðiÞ is the sum of degrees of vertices placed at the constant distance k from vertex i. It can be obtained by using a method devised by Lukovits [26]: X sk ðiÞ ¼ dðk kij Þd1 ðjÞ; ð4Þ
By combining Eqs. (4) and (5) and interchanging the order of summation we obtain: XX S‘ ¼ dðk kij Þd1 ðjÞ=2
j
where kij is the distance between vertices i and j, d1 ðjÞ is the degree of a vertex j and d is a binary function which is equal to unity if k ¼ kij and is zero in all other cases. The role of the binary function d is simply to ‘pick out’ all contributions at distance k from vertex i. If kij ¼ 0, then k ¼ 0, i.e., s0 ðiÞ is equal to the valence of vertex i itself, that is s0 ðiÞ ¼ d1 ðiÞ. The summation is over all vertices in a tree. The graph valence-shell Sk can be obtained by summing up expression (4) for vertices i and dividing the sum by two: X Sk ¼ sk ðiÞ=2: ð5Þ i
Note that in Randic’s notation [1], in contrast to our notation, sk ðiÞ denotes the valence-shell being at distance k–1 from vertex i. (Therefore in Randic’s notation s1 ðiÞ is the degree of vertex i.)
j
¼
X
i
dk ðjÞd1 ðjÞ:
ð6Þ
j
Eq. (6) is an alternative method to compute graph valence-shells. Fig. 3 serves as an illustration. For example, combining the second and fourth columns, i.e., d1 ðiÞ and d3 ðiÞ, we obtain: S3 ¼ ð1 3 þ 3 1 þ 4 0 þ 2 2 þ 1 3 þ 1 3 þ 1 3 þ 1 3Þ=2 ¼ 22=2 ¼ 11: The observation by Randic [1] that the combination of path-numbers P2 ; P3 ; P4 and graph valence-shells S2 and S3 , resulted in multivariate regressions with identical statistical characteristics is possible only if P2 ; P3 ; P4 and S2 ; S3 are linear transforms of each other. Specifically Randic found that: S2 ¼ P2 þ P3 ;
ð7Þ
I. Lukovits et al. / Chemical Physics Letters 354 (2002) 417–422
421
being at distance equal to two from vertex i. Therefore, d2 ðiÞ is the sum of vertex-degrees of all neighbors of vertex i minus all contributions originating from vertex i itself, i.e., d1 ðiÞ. From this it follows that s1 ðiÞ ¼ d1 ðiÞ þ d2 ðiÞ. The same argumentation can be used for case k > 1 and Eq. (11) can be obtained through induction. To summarize we point out that in addition to Eq. (9), graph valence-shell Sk may also be determined P by computing the sum of generalized degrees ½dk ðiÞ þ dkþ1 ðiÞ (Eq. (11)) and also by using Eq. (6).
Acknowledgements
Fig. 3. An illustrative example for Eq. (6).
S3 ¼ P 3 þ P 4 ;
ð8Þ
and concluded that: Sk ¼ Pk þ Pkþ1 :
ð9Þ
The latter equation was proved independently by Lukovits [26]. As a consequence of Eqs. (1), (3) and (9), we readily obtain: Sk ¼ ðDk þ Dkþ1 Þ=2:
ð10Þ
The authors are indebted to Dr. Christoph R€ ucker (Bayreuth) whose comments on early versions of this Letter has helped us to improve the presentation of this work. S.N. and N.T. were supported in part by the Ministry of Science and Technology of the Republic of Croatia via Grant No. 00980606. This work was also supported in part by the joint research program between the Croatian Academy of Sciences and Arts and the Hungarian Academy of Sciences.
Eq. (9) happens to be a special case of a more fundamental relationship: sk ðiÞ ¼ dk ðiÞ þ dkþ1 ðiÞ:
ð11Þ
From this, after summing for all vertices and applying the generalized handshaking lemma, Randic’s result (Eq. (9)) follows immediately. Illustrative example: obtain 2S3 by adding columns d3 ðiÞ and d4 ðiÞ in Fig. 3 2S3 ¼ bð3 þ 1Þ þ ð1 þ 0Þ þ ð0 þ 0Þ þ ð2 þ 0Þ þ ð3 þ 2Þ þ ð3 þ 1Þ þ ð3 þ 0Þ þ ð3 þ 0Þc ¼ 22; and therefore, S3 ¼ 11. Eq. (11) can be used to calculate valence-shells directly from generalized vertex-degrees. Proof of Eq. (11). ðk ¼ 1Þ consider vertex i. s1 ðiÞ is the sum of vertex-degrees being at distance 1 from vertex i. d2 ðiÞ is the second-order degree of vertex i, or in other words the number of vertices
References [1] M. Randic, J. Chem. Inf. Comput. Sci. 40 (2000) 627. [2] M.V. Diudea, J. Chem. Inf. Comput. Sci. 34 (1994) 1064. [3] M.V. Diudea, M. Topoan, A. Graovac, J. Chem. Inf. Comput. Sci. 34 (1994) 1072. [4] W.T. Tutte, Connectivity in Graphs, University of Toronto Press/Oxford University Press, London, 1966. [5] H. Hosoya, Bull. Chem. Soc. Jpn. 44 (1971) 2332. [6] J. Devillers, A.T. Balaban (Eds.), Topological Indices and Related Descriptors in QSAR and QSPR, Gordon and Breach, Amsterdam, 1999. [7] M. Karelson, Molecular Descriptors in QSAR/QSPR, Wiley–Interscience, New York, 2000. [8] R. Todeschini, V. Consonni, The Handbook of Molecular Descriptors, Wiley-VCH, New York, 2000. [9] M.V. Diudea (Ed.), QSPR/QSAR Studies by Molecular Descriptors, Nova Science Publishers, Hungtington, 2000. [10] I. Gutman, N. Trinajstic, Chem. Phys. Lett. 17 (1972) 535. [11] I. Gutman, B. Ruscic, N. Trinajstic, C.F. Wilcox Jr., J. Chem. Phys. 62 (1975) 3399. [12] S. Nikolic, I.M. Tolic, N. Trinajstic, I. Baucic, Croat. Chem. Acta 73 (2000) 909.
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I. Lukovits et al. / Chemical Physics Letters 354 (2002) 417–422
[13] M. Randic, J. Am. Chem. Soc. 97 (1975) 6609. [14] L. Pogliani, Chem. Rev. 100 (2000) 3827. [15] N. Trinajstic, Chemical Graph Theory, second revised edn., CRC Press, Boca Raton, FL, 1992. [16] I. Gutman, N. Trinajstic, Topics Curr. Chem. 42 (1973) 49. [17] A. Graovac, I. Gutman, N. Trinajstic, Topological Approach to the Chemistry of Conjugated Molecules, Springer, Berlin, 1977. [18] J.R. Platt, J. Phys. Chem. 56 (1952) 328. [19] H. Wiener, J. Am. Chem. Soc. 69 (1947) 17. ivkovic, Theor. [20] A. Graovac, I. Gutman, N. Trinajstic, T. Z Chim. Acta 26 (1972) 67.
[21] M. Randic, S.C. Basak, J. Chem. Inf. Comput. Sci. 39 (1999) 261. [22] M. Randic, M. Pompe, SAR QSAR Environ. Res. 10 (1999) 451. [23] D. Amic, S.C. Basak, B. Lucic, S. Nikolic, N. Trinajstic, Croat. Chem. Acta 74 (2001) 237. [24] L. Euler, Commentarii Academiae Scientarum Imperialis Petropolitanae 8 (1736) 128. [25] W.R. M€ uller, K. Szymanski, J.V. Knop, N. Trinajstic, J. Comput. Chem. 8 (1987) 170. [26] I. Lukovits, Commun. Math. Comput. Chem. (MATCH) 44 (2001) 279.
Chemical Physics Letters 354 (2002) 417–422 www.elsevier.com/locate/cplett
On relationships between vertex-degrees, path-numbers and graph valence-shells in trees Istv an Lukovits a, Sonja Nikolic b, Nenad Trinajstic a
b,*
Chemical Research Center, Hungarian Academy of Sciences, P.O. Box 17, H-1525 Budapest, Hungary b The Rugjer Boskovi c Institute, P.O. Box 180, HR-10002 Zagreb, Croatia Received 21 January 2002
Abstract It was shown that generalized vertex-degrees, path-numbers and graph valence-shells are closely related invariants in (chemical) trees. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction This Letter is motivated by a recent work of Randic [1] in which he has proposed a set of new graph invariants for chemical trees (which appears to be closely related to Diudea’s layer matrices [2,3]), calling them graph valence-shells. He has also shown that there exists a relationship between valence-shells and path-numbers. Our aim is to show that Randic’s result is a special case of a more general theorem.
2. Preliminaries The most elementary constituents of a (molecular) graph are vertices, edges, vertex-degrees, walks and paths [4]. They are the basis of many
*
Corresponding author. Fax: +385-1-468-0245. E-mail address: [email protected] (N. Trinajstic).
graph-theoretical invariants referred to (somewhat imprecisely) as topological indices [5], which have found considerable use in structure–property–activity modeling [6–9]. Zagreb indices [10–12] and the vertex-connectivity indices [13,14], which are given only in terms of vertex-degrees – although in different algebraic forms – are good examples. Structural formulas of molecules are converted into graphs simply by replacing words ‘atoms’ with ‘vertices’ and ‘bonds’ with ‘edges’ [15]. Trees are used to represent acyclic molecular structures. The degree dðiÞ of a vertex i (the vertex-degree) of a graph G is the number of adjacent vertices. However, we can redefine the vertex-degree dðiÞ by introducing the notion of the first neighbor [16,17]. All vertices among which the distance is unity (i.e., the adjacent vertices) are first neighbors. Thus, we can call dðiÞ the first-order vertex-degree, denote it by d1 ðiÞ and define it as the number of the first neighbors. The concept of neighbors allows the generalization of the concept of (first-order) vertex-degree to higher-order vertex-degrees. The
0009-2614/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 2 ) 0 0 1 5 3 - 7
418
I. Lukovits et al. / Chemical Physics Letters 354 (2002) 417–422
second-; third-; . . . ; kth-order vertex-degree, d2 ðiÞ; d3 ðiÞ; . . . dk ðiÞ, are the number of second; third; . . . ; kth neighbors, respectively. The hydrogen-depleted tree T representing the carbon skeleton of 2,3,3-trimethylpentane and the corresponding first-order and higher-order vertex-degrees are shown in Fig. 1. A path is a sequence of adjacent edges, which do not pass through the same vertex more than once [4]. The length of the path will be denoted by k. Path-number Pk denotes the number of paths of length k. Already almost fifty years ago Platt [18] advocated the use of path-numbers in structure– property modeling. However, this concept was already used by Wiener [19], who named the number of distances equal to three ‘polarity numbers’. However, the significance of Wiener’s and Platt’s
invention was overlooked until resurrection of chemical graph theory in the early seventies [5,20]. Recently several papers appeared in which quantitative structure–property relationships (QSPR) modeling was performed using path-numbers [21– 23]. In most cases QSPR models based on pathnumbers possessed better statistical parameters as compared to models based on other kinds of molecular descriptors. Therefore, path-numbers appear to be molecular descriptors that may play important role both in QSPR and quantitative structure–activity relationships (QSAR) modeling. This fact alone warrants the current interests in the properties of these numbers [1–3]. In Fig. 2 we give path-numbers for the 2,3,3-trimethylpentane tree shown in Fig. 1. Note that path-number P1 is equal to the number of edges E in a tree.
Fig. 1. Tree T representing the carbon skeleton of 2,3,3-trimethylpentane and the corresponding first- and higher-order vertex-degrees.
I. Lukovits et al. / Chemical Physics Letters 354 (2002) 417–422
419
Fig. 2. Path-numbers for the 2,3,3-trimethylpentane tree. Thick lines denote paths.
3. Vertex-degrees and path-numbers There is a simple relationship between vertexdegrees and path-numbers, which is obvious from Figs. 1 and 2: X Pk ¼ ð1=2Þ dk ðiÞ: ð1Þ i
This relationship is valid only for trees. Eq. (1) represents a generalization of the handshaking lemma, proved by Euler [24] in 1736: X 2E ¼ d1 ðiÞ: ð2Þ i
A consequence of the close relationship between vertex-degrees and path-numbers is that first- and higher-order vertex-degrees may replace pathnumbers in QSPR models for acyclic molecules with no effect on statistical parameters of the model as shown by Randic [1].
Both vertex-degrees and path-numbers can be straightforwardly obtained from the distance matrix [15]. A very efficient algorithm, developed by M€ uller et al. [25], is available for computing the distance matrix of any graph. Once we possess the distance matrix of a tree, we can easily obtain the total vertex-degree of the kth order, Dk , which is equal to the number of distances of the k-length. Path-numbers can be obtained in the similar way from the upper (or the lower) triangle of the distance matrix: a given path-number Pk is equal to the number of distances of length k in the triangle. The correctness of the calculated path-numbers can simply be checked by using Eq. (1). It should also be noted that the total vertex-degree of order k is given by expression: X Dk ¼ dk ðiÞ: ð3Þ i
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I. Lukovits et al. / Chemical Physics Letters 354 (2002) 417–422
Table 1 The distance matrix and the corresponding total vertex-degrees and the path-numbers of the 2,3,3-trimethylpentane tree shown in Fig. 1 0 1 2 3 4 2 3 3
1 0 1 2 3 1 2 2
2 1 0 1 2 2 1 1 D1 D2 D3 D4
3 2 1 0 1 3 2 2 ¼ 14 ¼ 20 ¼ 18 ¼4
4 3 2 1 0 4 3 3 P1 P2 P3 P4
2 1 2 3 4 0 3 3
3 2 1 2 3 3 0 2
3 2 1 2 3 3 2 0
¼7 ¼ 10 ¼9 ¼2
In Table 1 we give the distance matrix and the corresponding total vertex-degrees and pathnumbers of the 2,3,3-trimethylpentane tree shown in Fig. 1.
The calculation of Sk for given k can be illustrated by using as an example the 2,3,3-trimethylpentane tree shown in Fig. 1. For example, the value of the S3 can be obtained as follows: (i) the valence-shell of vertex 1 is 4 (sum of degrees of vertices 4, 7 and 8), (ii) the valence-shell of vertex 2 is 1 (degree of vertex 5), (iii) the valence-shell of vertex 3 is 0 (there is no vertex at distance equal to three from vertex 3), (iv) the valence-shell of vertex 4 is 2 (sum of degrees of vertices 1 and 6), (v) the valence-shell of vertex 5 is 5 (sum of degrees of vertices 2, 7 and 8), (vi) the valence-shell of vertex 6 is 4 (sum of degrees of vertices 4, 7 and 8), (vii) the valence-shell of vertex 7 is 3 (sum of degrees of vertices 1, 5 and 6), (viii) the valence-shell of vertex 8 is 3 (sum of degrees of vertices 1, 5 and 6). Therefore,
3.1. Generalization of Randi c’s result
S3 ¼ ð4 þ 1 þ 0 þ 2 þ 5 þ 4 þ 3 þ 3Þ=2 ¼ 11:
A valence-shell sk ðiÞ is the sum of degrees of vertices placed at the constant distance k from vertex i. It can be obtained by using a method devised by Lukovits [26]: X sk ðiÞ ¼ dðk kij Þd1 ðjÞ; ð4Þ
By combining Eqs. (4) and (5) and interchanging the order of summation we obtain: XX S‘ ¼ dðk kij Þd1 ðjÞ=2
j
where kij is the distance between vertices i and j, d1 ðjÞ is the degree of a vertex j and d is a binary function which is equal to unity if k ¼ kij and is zero in all other cases. The role of the binary function d is simply to ‘pick out’ all contributions at distance k from vertex i. If kij ¼ 0, then k ¼ 0, i.e., s0 ðiÞ is equal to the valence of vertex i itself, that is s0 ðiÞ ¼ d1 ðiÞ. The summation is over all vertices in a tree. The graph valence-shell Sk can be obtained by summing up expression (4) for vertices i and dividing the sum by two: X Sk ¼ sk ðiÞ=2: ð5Þ i
Note that in Randic’s notation [1], in contrast to our notation, sk ðiÞ denotes the valence-shell being at distance k–1 from vertex i. (Therefore in Randic’s notation s1 ðiÞ is the degree of vertex i.)
j
¼
X
i
dk ðjÞd1 ðjÞ:
ð6Þ
j
Eq. (6) is an alternative method to compute graph valence-shells. Fig. 3 serves as an illustration. For example, combining the second and fourth columns, i.e., d1 ðiÞ and d3 ðiÞ, we obtain: S3 ¼ ð1 3 þ 3 1 þ 4 0 þ 2 2 þ 1 3 þ 1 3 þ 1 3 þ 1 3Þ=2 ¼ 22=2 ¼ 11: The observation by Randic [1] that the combination of path-numbers P2 ; P3 ; P4 and graph valence-shells S2 and S3 , resulted in multivariate regressions with identical statistical characteristics is possible only if P2 ; P3 ; P4 and S2 ; S3 are linear transforms of each other. Specifically Randic found that: S2 ¼ P2 þ P3 ;
ð7Þ
I. Lukovits et al. / Chemical Physics Letters 354 (2002) 417–422
421
being at distance equal to two from vertex i. Therefore, d2 ðiÞ is the sum of vertex-degrees of all neighbors of vertex i minus all contributions originating from vertex i itself, i.e., d1 ðiÞ. From this it follows that s1 ðiÞ ¼ d1 ðiÞ þ d2 ðiÞ. The same argumentation can be used for case k > 1 and Eq. (11) can be obtained through induction. To summarize we point out that in addition to Eq. (9), graph valence-shell Sk may also be determined P by computing the sum of generalized degrees ½dk ðiÞ þ dkþ1 ðiÞ (Eq. (11)) and also by using Eq. (6).
Acknowledgements
Fig. 3. An illustrative example for Eq. (6).
S3 ¼ P 3 þ P 4 ;
ð8Þ
and concluded that: Sk ¼ Pk þ Pkþ1 :
ð9Þ
The latter equation was proved independently by Lukovits [26]. As a consequence of Eqs. (1), (3) and (9), we readily obtain: Sk ¼ ðDk þ Dkþ1 Þ=2:
ð10Þ
The authors are indebted to Dr. Christoph R€ ucker (Bayreuth) whose comments on early versions of this Letter has helped us to improve the presentation of this work. S.N. and N.T. were supported in part by the Ministry of Science and Technology of the Republic of Croatia via Grant No. 00980606. This work was also supported in part by the joint research program between the Croatian Academy of Sciences and Arts and the Hungarian Academy of Sciences.
Eq. (9) happens to be a special case of a more fundamental relationship: sk ðiÞ ¼ dk ðiÞ þ dkþ1 ðiÞ:
ð11Þ
From this, after summing for all vertices and applying the generalized handshaking lemma, Randic’s result (Eq. (9)) follows immediately. Illustrative example: obtain 2S3 by adding columns d3 ðiÞ and d4 ðiÞ in Fig. 3 2S3 ¼ bð3 þ 1Þ þ ð1 þ 0Þ þ ð0 þ 0Þ þ ð2 þ 0Þ þ ð3 þ 2Þ þ ð3 þ 1Þ þ ð3 þ 0Þ þ ð3 þ 0Þc ¼ 22; and therefore, S3 ¼ 11. Eq. (11) can be used to calculate valence-shells directly from generalized vertex-degrees. Proof of Eq. (11). ðk ¼ 1Þ consider vertex i. s1 ðiÞ is the sum of vertex-degrees being at distance 1 from vertex i. d2 ðiÞ is the second-order degree of vertex i, or in other words the number of vertices
References [1] M. Randic, J. Chem. Inf. Comput. Sci. 40 (2000) 627. [2] M.V. Diudea, J. Chem. Inf. Comput. Sci. 34 (1994) 1064. [3] M.V. Diudea, M. Topoan, A. Graovac, J. Chem. Inf. Comput. Sci. 34 (1994) 1072. [4] W.T. Tutte, Connectivity in Graphs, University of Toronto Press/Oxford University Press, London, 1966. [5] H. Hosoya, Bull. Chem. Soc. Jpn. 44 (1971) 2332. [6] J. Devillers, A.T. Balaban (Eds.), Topological Indices and Related Descriptors in QSAR and QSPR, Gordon and Breach, Amsterdam, 1999. [7] M. Karelson, Molecular Descriptors in QSAR/QSPR, Wiley–Interscience, New York, 2000. [8] R. Todeschini, V. Consonni, The Handbook of Molecular Descriptors, Wiley-VCH, New York, 2000. [9] M.V. Diudea (Ed.), QSPR/QSAR Studies by Molecular Descriptors, Nova Science Publishers, Hungtington, 2000. [10] I. Gutman, N. Trinajstic, Chem. Phys. Lett. 17 (1972) 535. [11] I. Gutman, B. Ruscic, N. Trinajstic, C.F. Wilcox Jr., J. Chem. Phys. 62 (1975) 3399. [12] S. Nikolic, I.M. Tolic, N. Trinajstic, I. Baucic, Croat. Chem. Acta 73 (2000) 909.
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