The concept of thorn graphs proposed recently by Gutman is extended to the broader concept of generalized thorny graphs. The latter preserve unchanged certain parts of the parent graph while applying the procedure of thorn graph generation to the rest of the graph. Thorn stars and rods are typical examples of such generalized thorn graphs, owing to the unchanged star center, and the other-than-terminal rod vertices. Formulae are presented for the Wiener number of these and other classes of (generalized) thorn graphs

Danail Bonchev

Douglas J. Klein

Croatica Chemica Acta

English

Hrčak - Portal of scientific journals of Croatia

ISSN-0011-1643CCA-2820 Original Scientific PaperOn the Wiener Number of Thorn Trees,Stars, Rings, and Rods,#Danail Bonchev** and Douglas J. KleinTexas A&M University, Fort Crockett Campus, 5007 Avenue U,Galveston, TX 77551, USAReceived June 5, 2001; revised February 21, 2002; accepted February 22, 2002The concept of thorn graphs proposed recently by Gutman is ex-tended to the broader concept of generalized thorny graphs. Thelatter preserve unchanged certain parts of the parent graph whileapplying the procedure of thorn graph generation to the rest of thegraph. Thorn stars and rods are typical examples of such generali-zed thorn graphs, owing to the unchanged star center, and the other-than-terminal rod vertices. Formulae are presented for the Wienernumber of these and other classes of (generalized) thorn graphs.Key words: (generalized) thorn graphs, Wiener number, thorn rings,thorn stars, thorn rods.INTRODUCTIONThe Wiener number1 is one of the most common topological descriptorsof molecular structure, an area of theoretical chemistry to which Milan Ran-di} made significant contributions.2,3Recently, Gutman4,5 introduced the concept of »thorn graph« for anygraph G* that can be obtained from a parent connected graph G by attach-ing pi  0 new vertices of degree one to each vertex i. Theorems have beenderived for a special case of thorn graphs for which pi = amax – ai, where amaxis a constant, and ai is the degree of the i-th vertex in G, G being a tree. ForDedicated to Professor Milan Randi} on the occasion of his 70th birthday.#Reported in part on the Second Indo-American Workshop on Mathematical Chemistry,May 30 – June 3, 2000, University of Minnesota, Duluth, Minnesota, USA.** Author to whom correspondence should be addressed. (E-mail: bonchev@aol.com)CROATICA CHEMICA ACTA CCACAA 75 (2) 613¿620 (2002)organic acyclic molecules, amax = 4 or 3 for the respective classes of alkanes(as in the transformation of 1) or polyenoids (as in 2).These classes of compounds have only non-terminal vertices of degree t= amax and terminal vertices of degree 1. Such graphs are sometimes calledproper graphs.6 One might thus regard proper graphs as a special case ofthorny graphs with a uniform vertex degree t of all non-terminal vertices.We will call such graphs t-thorn graphs. The t-thorn graphs are of impor-tance for polymer theory, especially for dendrimers.7 The Wiener number ofpolymer graphs could be of use in the quantitative characterization of poly-mer topology,8,9 as well as for specific structure-property relationships. For-mulae for the Wiener number of dendrimer species have been reported byGutman10 and Diudea.11 Recently, Dobrynin, Entringer, and Gutman12 re-viewed the studies on the Wiener index of trees.Gutman4 derived a formula that relates the Wiener number W of suchthorn trees T* with the Wiener number of the parent graph T having n ver-tices:W(T*) = (t–1)2W(T) + (t–1)n+12 . (1)Bytautas, Bonchev, and Klein13 calculated the average Wiener numberfor the (structural) isomer classes of alkanes and polyenoids having up to100 carbon atoms, making use of formula (1) and a generation-series-basedmethodology for calculating average properties of isomer series.14–15 In thispaper we present closed-form formulae for calculating the Wiener number ofseveral classes of thorn graphs, such as caterpillars having a chain as a par-ent graph, and thorn rings whose parent graph is a monocycle. We also ap-ply the notion of generalized thorn graphs, to consider stars and rods. Suchgraphs preserve the vertex degrees in some essential structural fragmentsof the parent graph such as the central vertex of a star or the internal partof a chain, whereas the procedure of the thorn graph formation is applied to614 D. BONCHEV AND D. J. KLEIN12the rest of the graph vertices. The Wiener number formulae for these twoclasses of general graphs are also presented. The formulae are given in aform containing essential structural information; besides the number ofthorns these are the rod length and the number of star arms.THE WIENER NUMBER FORMULAE FOR THORN TREESCaterpillarsDefinition: A caterpillar is a thorn tree T*(a,b) whose nonterminal verticesb  1 are of the same degree a > 2, and whose parent graph is the b-sitechain graph, Pb.Theorem: The Wiener number of a caterpillar having b branched vertices ofdegree a is:W(T*(a,b)) =16b(a–1)(a–1)(b–1)(b+7) + 6(a+1) + 1 . (2)Proof: We apply the formula originally used by Wiener:W N Ni j 12 all edges(3)where Ni and Nj are the numbers of vertices in the left and right compo-nents to which the graph is decomposed after cutting a graph edge i, j, andthe sum of Eq. (3) is over all edges in the graph. The total number of verti-ces in T*(a,b) isn = (a–1)b + 2 . (4)Applying the Wiener constructive procedure one obtainsW = 1  (n–1)  (a–2)b+2 + a  (n–a) + (2a–1)  n–(2a–1) +(3a–2)  n–(3a–2) + ... + 	b–1)a – (b–2)  n – (b–1)a + (b–2) =(a–1)b+1(a–2)b+2 +ib11i(a–1) + 1(a–1)(b–i) + 1WIENER NUMBER OF THORN GRAPHS 615a b= 3, = 6 a b= 4, = 7wherefrom one arrives after some elementary transformations at formula(2). For the chemically relevant cases of a = 3 and a = 4 one obtains the Wie-ner number as a polynomials of degree three in the number of branched ver-tices:W(b,a = 3) =13(2b3 + 12b2 + 10b + 3) (2a)W(b,a = 4) =12(3b3 + 18b2 + 9b + 2) . (2b)It can be shown that our formula (2) along with the formula for W(Pb) isa consequence of Gutman’s formula (1).Thorny RingsDefinition: A t-thorny ring has a simple cycle as the parent, and t-2 thornsat each cycle vertex.Theorem: The Wiener number of a t-thorny ring having n ring-vertices isW =18n(t–1)2 (n2–a) + n(t–2)(nt–n–1) (5)with a = 0 for n – even, and a = 1 for n – odd.Proof: Due to symmetry of the t-thorn ring, the Wiener number may be ex-pressed by the sum d1 of the distances to a ring vertex 1 and that d2 to a ter-minal vertex 2:W(Cn,t) =12nd1 + (t–2)d2 . (6)By induction, one hasd2 = d1 + (t–1)n–2 (7)616 D. BONCHEV AND D. J. KLEINn t= 3, = 3 n t= 4, = 3 n t= 5, = 3 n t= 6, = 3d1(n–even) =14(t–1)n2 + 4(t–2)n;d1(n–odd) =14(t–1)n2 + 4(t–2)n – t +1 . (8)Substituting (7) and (8) into (6) one obtains Eq. (5).Substituting in (5) n3/8 = Cn-even and (n3–n)/8 = Cn-odd, one can expressthe Wiener number of a t-thorn ring, W(Cn,t), by the Wiener number of theparent cyclic graph, W(Cn):W(Cn,t) = (t–1)2 W(Cn) + n(t–2) (nt – n – 1) . (9)Eq. (9) for the thorn rings coincides with the equation given as Corollary1.1 by Gutman4 for the general case of thorn graphs G* obtained from a par-ent graph G by attaching the same number of vertices t at each of the verti-ces of G.Thorn RodsDefinition: A thorn rod is a graph, Pp,t, which includes a linear chain (ter-med »rod«) of p vertices and degree-t terminal vertices at each of the two rodends.Theorem: The Wiener number of a thorn rod Pp,t isW(Pp,t) =16(p3–p) + (t–1)2 (p+3) + (t–1)(p2+p–2) . (10)Proof: We proceed from the Doyle-Graver formula16Wnn n ni j ki j ku  13. (11)Here, n is the total number of vertices in the graph, and ni, nj, and nkare the numbers of vertices in the branches i, j, and k emanating from ver-tex u, and the first summation is over all branched vertices. The thorn rodWIENER NUMBER OF THORN GRAPHS 617p t= 6, = 3 p t= 7, = 4has only two branched vertices each one having t–1 branches of a single ver-tex and a large branch with p + t – 2 vertices. Then, from (11) one getsW(Pp,t) =n t t  13213212(p + t – 2) . (12)After substituting n = p + 2t – 2 and simplifying Eq. (12) one arrives atEq. (10). The specific case with p = 2, corresponds to the »pom-pom« model,widely used in polymer theory:W(P2,t) = 5(t–1)2 + 4(t–1) + 1 . (16a)Corollary: The Wiener number of the t-thorn rod can be obtained fromthe Wiener number of the parent path graph, W(Pp), by the formula:W(Pp,t) = W(Pp) + (t–1)2(p+3) + (t–1)(p2+p–2) . (17)The proof follows directly from Eq. (16) the first term in which is exactlythe Wiener number of the parent path graph.Thorn StarsDefinition: Thorn stars are graphs obtained from a k-arm star by attachingt-1 terminal vertices to each of the star arms.Theorem: The Wiener number of a thorn star Sk,t isW(Sk,t) = kt(2kt–k–t+1) (18)Proof: We proceed again from the Doyle-Graver formula (11) for the Wienernumber of a branched tree. There are k+1 branched vertices in Sk,t, k ofwhich are equivalent and have t–1 branches of one vertex and one branch of618 D. BONCHEV AND D. J. KLEINk t= 3, = 4 k t= 4, = 3n-t vertices. The central vertex has k equivalent branches, each one with tvertices. Therefore, (11) transforms intoW =nktn tt   131312( ) kt33 . (19)Expressing the total number of vertices in Sk,t asn = kt + 1 (20)and substituting them in (19), one proves the theorem.Eq. (20) shows that for a given star, the Wiener number is a quadraticfunction of the number of thorns t. For the chemically relevant cases of 3-arm star and 4-arm star one obtainsW(S3,t) = 15(t–1)2 + 24(t–1) + 9 (18a)W(S4,t) = 28(t–1)2 + 44(t–1) + 16 (18b)Corollary: The Wiener number of the t-thorn star, W(Sk,t), is related tothe Wiener number of the parent star, W(Sk), by the equationW(Sk,t) = W(Sk) + k(t–1)2k(t–1) + 3k–t (21)The proof follows directly from comparing Eq. (18) to the equation17W(Sk) = k2 . (22)REFERENCES1. (a) H. Wiener, J. Am. Chem. Soc. 69 (1947) 17–20; (b) J. Phys. Chem. 52 (1948)1082–1089.2. M. Randi}, J. Chem. Inf. Comput. Sci. 33 (1993) 709–716.3. M. Randi}, J. Chem. Inf. Comput. Sci. 34 (1994) 361–367.4. I. Gutman, Publ. Inst. Math. (Beograd) 63 (1998) 31–36.5. I. Gutman, D. Vidovi}, and L. Popovi}, J. Chem. Soc., Faraday Trans. 94 (1998)857–860.6. F. Harary, Graph Theory, 2nded., Addison-Wesley, Reading, MA, 1969.7. M. V. Diudea and G. Katona, Adv. Dendritic Macromol. 4 (1999) 135–201.8. D. Bonchev, E. Markel, and A. Dekmezian, Polymer 43 (2002) 203–222.9. D. Bonchev, E. Markel, and A. Dekmezian, J. Chem. Inf. Comput. Sci. 41 (2001)1274–1285.10. I. Gutman, Y. N. Yeh, S. L. Lee, and J. C. Chen, MATCH (Commun. Math. Com-put. Chem.) 30 (1994) 103–115.WIENER NUMBER OF THORN GRAPHS 61911. M. W. Diudea, MATCH (Commun. Math. Comput. Chem.) 35 (1995) 71–83.12. A. A. Dobrynin, R. Entringer, and I. Gutman, Acta Appl. Math. 66 (2001) 211–249.13. L. Bytautas, D. Bonchev, and D. J. Klein, MATCH (Commun. Math. Comput.Chem.), in press.14. L. Bytautas and D. J. Klein, J. Chem. Inf. Comput. Sci. 38 (1998) 1063–1078.15. L. Bytautas and D. J. Klein, J. Chem. Inf. Comput. Sci. 40 (2000) 471–481.16. J. K. Doyle and J. E. Graver, Discrete Math. 17 (1977) 147–154.17. D. Bonchev and N. Trinajsti}, J. Chem. Phys. 67 (1977) 4517–4533.SA@ETAKO Wienerovu broju trnovitih stabala, zvijezda, prstenova i prutovaDanail Bonchev i Douglas J. KleinGraf-teorijska koncepcija o trnovitim stablima poop}ena je u koncepciju o trno-vitim grafovima. Ona dopu{ta da izvjesni dijelovi izvornoga grafa ostanu nepromije-njeni prilikom primjene postupka generiranja trnovitih grafova na ostatak izvornogagrafa. Trnovite zvijezde i prutovi tipi~ni su primjeri poop}enih trnovitih grafova snepromijenjenim sredi{tem zvijezde i nedirnutim unutarnjim ~vorovima pruta. Danesu formule za izra~unavanje Wienerova broja tih i drugih klasa (poop}enih) trnovitihgrafova.620 D. BONCHEV AND D. J. KLEIN

On the Wiener Number of Thorn Trees, Stars, Rings, and Rods

Bonchev, Danail

Klein, Douglas J

Texas A&amp;M Repository

ISSN-0011-1643CCA-2820 Original Scientific PaperOn the Wiener Number of Thorn Trees,Stars, Rings, and Rods,#Danail Bonchev** and Douglas J. KleinTexas A&M University, Fort Crockett Campus, 5007 Avenue U,Galveston, TX 77551, USAReceived June 5, 2001; revised February 21, 2002; accepted February 22, 2002The concept of thorn graphs proposed recently by Gutman is ex-tended to the broader concept of generalized thorny graphs. Thelatter preserve unchanged certain parts of the parent graph whileapplying the procedure of thorn graph generation to the rest of thegraph. Thorn stars and rods are typical examples of such generali-zed thorn graphs, owing to the unchanged star center, and the other-than-terminal rod vertices. Formulae are presented for the Wienernumber of these and other classes of (generalized) thorn graphs.Key words: (generalized) thorn graphs, Wiener number, thorn rings,thorn stars, thorn rods.INTRODUCTIONThe Wiener number1 is one of the most common topological descriptorsof molecular structure, an area of theoretical chemistry to which Milan Ran-di} made significant contributions.2,3Recently, Gutman4,5 introduced the concept of »thorn graph« for anygraph G* that can be obtained from a parent connected graph G by attach-ing pi  0 new vertices of degree one to each vertex i. Theorems have beenderived for a special case of thorn graphs for which pi = amax – ai, where amaxis a constant, and ai is the degree of the i-th vertex in G, G being a tree. ForDedicated to Professor Milan Randi} on the occasion of his 70th birthday.#Reported in part on the Second Indo-American Workshop on Mathematical Chemistry,May 30 – June 3, 2000, University of Minnesota, Duluth, Minnesota, USA.** Author to whom correspondence should be addressed. (E-mail: bonchev@aol.com)CROATICA CHEMICA ACTA CCACAA 75 (2) 613¿620 (2002)organic acyclic molecules, amax = 4 or 3 for the respective classes of alkanes(as in the transformation of 1) or polyenoids (as in 2).These classes of compounds have only non-terminal vertices of degree t= amax and terminal vertices of degree 1. Such graphs are sometimes calledproper graphs.6 One might thus regard proper graphs as a special case ofthorny graphs with a uniform vertex degree t of all non-terminal vertices.We will call such graphs t-thorn graphs. The t-thorn graphs are of impor-tance for polymer theory, especially for dendrimers.7 The Wiener number ofpolymer graphs could be of use in the quantitative characterization of poly-mer topology,8,9 as well as for specific structure-property relationships. For-mulae for the Wiener number of dendrimer species have been reported byGutman10 and Diudea.11 Recently, Dobrynin, Entringer, and Gutman12 re-viewed the studies on the Wiener index of trees.Gutman4 derived a formula that relates the Wiener number W of suchthorn trees T* with the Wiener number of the parent graph T having n ver-tices:W(T*) = (t–1)2W(T) + (t–1)n+12 . (1)Bytautas, Bonchev, and Klein13 calculated the average Wiener numberfor the (structural) isomer classes of alkanes and polyenoids having up to100 carbon atoms, making use of formula (1) and a generation-series-basedmethodology for calculating average properties of isomer series.14–15 In thispaper we present closed-form formulae for calculating the Wiener number ofseveral classes of thorn graphs, such as caterpillars having a chain as a par-ent graph, and thorn rings whose parent graph is a monocycle. We also ap-ply the notion of generalized thorn graphs, to consider stars and rods. Suchgraphs preserve the vertex degrees in some essential structural fragmentsof the parent graph such as the central vertex of a star or the internal partof a chain, whereas the procedure of the thorn graph formation is applied to614 D. BONCHEV AND D. J. KLEIN12the rest of the graph vertices. The Wiener number formulae for these twoclasses of general graphs are also presented. The formulae are given in aform containing essential structural information; besides the number ofthorns these are the rod length and the number of star arms.THE WIENER NUMBER FORMULAE FOR THORN TREESCaterpillarsDefinition: A caterpillar is a thorn tree T*(a,b) whose nonterminal verticesb  1 are of the same degree a > 2, and whose parent graph is the b-sitechain graph, Pb.Theorem: The Wiener number of a caterpillar having b branched vertices ofdegree a is:W(T*(a,b)) =16b(a–1)(a–1)(b–1)(b+7) + 6(a+1) + 1 . (2)Proof: We apply the formula originally used by Wiener:W N Ni j 12 all edges(3)where Ni and Nj are the numbers of vertices in the left and right compo-nents to which the graph is decomposed after cutting a graph edge i, j, andthe sum of Eq. (3) is over all edges in the graph. The total number of verti-ces in T*(a,b) isn = (a–1)b + 2 . (4)Applying the Wiener constructive procedure one obtainsW = 1  (n–1)  (a–2)b+2 + a  (n–a) + (2a–1)  n–(2a–1) +(3a–2)  n–(3a–2) + ... + 	b–1)a – (b–2)  n – (b–1)a + (b–2) =(a–1)b+1(a–2)b+2 +ib11i(a–1) + 1(a–1)(b–i) + 1WIENER NUMBER OF THORN GRAPHS 615a b= 3, = 6 a b= 4, = 7wherefrom one arrives after some elementary transformations at formula(2). For the chemically relevant cases of a = 3 and a = 4 one obtains the Wie-ner number as a polynomials of degree three in the number of branched ver-tices:W(b,a = 3) =13(2b3 + 12b2 + 10b + 3) (2a)W(b,a = 4) =12(3b3 + 18b2 + 9b + 2) . (2b)It can be shown that our formula (2) along with the formula for W(Pb) isa consequence of Gutman’s formula (1).Thorny RingsDefinition: A t-thorny ring has a simple cycle as the parent, and t-2 thornsat each cycle vertex.Theorem: The Wiener number of a t-thorny ring having n ring-vertices isW =18n(t–1)2 (n2–a) + n(t–2)(nt–n–1) (5)with a = 0 for n – even, and a = 1 for n – odd.Proof: Due to symmetry of the t-thorn ring, the Wiener number may be ex-pressed by the sum d1 of the distances to a ring vertex 1 and that d2 to a ter-minal vertex 2:W(Cn,t) =12nd1 + (t–2)d2 . (6)By induction, one hasd2 = d1 + (t–1)n–2 (7)616 D. BONCHEV AND D. J. KLEINn t= 3, = 3 n t= 4, = 3 n t= 5, = 3 n t= 6, = 3d1(n–even) =14(t–1)n2 + 4(t–2)n;d1(n–odd) =14(t–1)n2 + 4(t–2)n – t +1 . (8)Substituting (7) and (8) into (6) one obtains Eq. (5).Substituting in (5) n3/8 = Cn-even and (n3–n)/8 = Cn-odd, one can expressthe Wiener number of a t-thorn ring, W(Cn,t), by the Wiener number of theparent cyclic graph, W(Cn):W(Cn,t) = (t–1)2 W(Cn) + n(t–2) (nt – n – 1) . (9)Eq. (9) for the thorn rings coincides with the equation given as Corollary1.1 by Gutman4 for the general case of thorn graphs G* obtained from a par-ent graph G by attaching the same number of vertices t at each of the verti-ces of G.Thorn RodsDefinition: A thorn rod is a graph, Pp,t, which includes a linear chain (ter-med »rod«) of p vertices and degree-t terminal vertices at each of the two rodends.Theorem: The Wiener number of a thorn rod Pp,t isW(Pp,t) =16(p3–p) + (t–1)2 (p+3) + (t–1)(p2+p–2) . (10)Proof: We proceed from the Doyle-Graver formula16Wnn n ni j ki j ku 13. (11)Here, n is the total number of vertices in the graph, and ni, nj, and nkare the numbers of vertices in the branches i, j, and k emanating from ver-tex u, and the first summation is over all branched vertices. The thorn rodWIENER NUMBER OF THORN GRAPHS 617p t= 6, = 3 p t= 7, = 4has only two branched vertices each one having t–1 branches of a single ver-tex and a large branch with p + t – 2 vertices. Then, from (11) one getsW(Pp,t) =n t t13213212(p + t – 2) . (12)After substituting n = p + 2t – 2 and simplifying Eq. (12) one arrives atEq. (10). The specific case with p = 2, corresponds to the »pom-pom« model,widely used in polymer theory:W(P2,t) = 5(t–1)2 + 4(t–1) + 1 . (16a)Corollary: The Wiener number of the t-thorn rod can be obtained fromthe Wiener number of the parent path graph, W(Pp), by the formula:W(Pp,t) = W(Pp) + (t–1)2(p+3) + (t–1)(p2+p–2) . (17)The proof follows directly from Eq. (16) the first term in which is exactlythe Wiener number of the parent path graph.Thorn StarsDefinition: Thorn stars are graphs obtained from a k-arm star by attachingt-1 terminal vertices to each of the star arms.Theorem: The Wiener number of a thorn star Sk,t isW(Sk,t) = kt(2kt–k–t+1) (18)Proof: We proceed again from the Doyle-Graver formula (11) for the Wienernumber of a branched tree. There are k+1 branched vertices in Sk,t, k ofwhich are equivalent and have t–1 branches of one vertex and one branch of618 D. BONCHEV AND D. J. KLEINk t= 3, = 4 k t= 4, = 3n-t vertices. The central vertex has k equivalent branches, each one with tvertices. Therefore, (11) transforms intoW =nktn tt 131312( ) kt33 . (19)Expressing the total number of vertices in Sk,t asn = kt + 1 (20)and substituting them in (19), one proves the theorem.Eq. (20) shows that for a given star, the Wiener number is a quadraticfunction of the number of thorns t. For the chemically relevant cases of 3-arm star and 4-arm star one obtainsW(S3,t) = 15(t–1)2 + 24(t–1) + 9 (18a)W(S4,t) = 28(t–1)2 + 44(t–1) + 16 (18b)Corollary: The Wiener number of the t-thorn star, W(Sk,t), is related tothe Wiener number of the parent star, W(Sk), by the equationW(Sk,t) = W(Sk) + k(t–1)2k(t–1) + 3k–t (21)The proof follows directly from comparing Eq. (18) to the equation17W(Sk) = k2 . (22)REFERENCES1. (a) H. Wiener, J. Am. Chem. Soc. 69 (1947) 17–20; (b) J. Phys. Chem. 52 (1948)1082–1089.2. M. Randi}, J. Chem. Inf. Comput. Sci. 33 (1993) 709–716.3. M. Randi}, J. Chem. Inf. Comput. Sci. 34 (1994) 361–367.4. I. Gutman, Publ. Inst. Math. (Beograd) 63 (1998) 31–36.5. I. Gutman, D. Vidovi}, and L. Popovi}, J. Chem. Soc., Faraday Trans. 94 (1998)857–860.6. F. Harary, Graph Theory, 2nded., Addison-Wesley, Reading, MA, 1969.7. M. V. Diudea and G. Katona, Adv. Dendritic Macromol. 4 (1999) 135–201.8. D. Bonchev, E. Markel, and A. Dekmezian, Polymer 43 (2002) 203–222.9. D. Bonchev, E. Markel, and A. Dekmezian, J. Chem. Inf. Comput. Sci. 41 (2001)1274–1285.10. I. Gutman, Y. N. Yeh, S. L. Lee, and J. C. Chen, MATCH (Commun. Math. Com-put. Chem.) 30 (1994) 103–115.WIENER NUMBER OF THORN GRAPHS 61911. M. W. Diudea, MATCH (Commun. Math. Comput. Chem.) 35 (1995) 71–83.12. A. A. Dobrynin, R. Entringer, and I. Gutman, Acta Appl. Math. 66 (2001) 211–249.13. L. Bytautas, D. Bonchev, and D. J. Klein, MATCH (Commun. Math. Comput.Chem.), in press.14. L. Bytautas and D. J. Klein, J. Chem. Inf. Comput. Sci. 38 (1998) 1063–1078.15. L. Bytautas and D. J. Klein, J. Chem. Inf. Comput. Sci. 40 (2000) 471–481.16. J. K. Doyle and J. E. Graver, Discrete Math. 17 (1977) 147–154.17. D. Bonchev and N. Trinajsti}, J. Chem. Phys. 67 (1977) 4517–4533.SA@ETAKO Wienerovu broju trnovitih stabala, zvijezda, prstenova i prutovaDanail Bonchev i Douglas J. KleinGraf-teorijska koncepcija o trnovitim stablima poop}ena je u koncepciju o trno-vitim grafovima. Ona dopu{ta da izvjesni dijelovi izvornoga grafa ostanu nepromije-njeni prilikom primjene postupka generiranja trnovitih grafova na ostatak izvornogagrafa. Trnovite zvijezde i prutovi tipi~ni su primjeri poop}enih trnovitih grafova snepromijenjenim sredi{tem zvijezde i nedirnutim unutarnjim ~vorovima pruta. Danesu formule za izra~unavanje Wienerova broja tih i drugih klasa (poop}enih) trnovitihgrafova.620 D. BONCHEV AND D. J. KLEIN

Texas A&M University

ISSN-0011-1643
CCA-2820 Original Scientific Paper
On the Wiener Number of Thorn Trees,
Stars, Rings, and Rods,#
Danail Bonchev** and Douglas J. Klein
Texas A&M University, Fort Crockett Campus, 5007 Avenue U,
Galveston, TX 77551, USA
Received June 5, 2001; revised February 21, 2002; accepted February 22, 2002
The concept of thorn graphs proposed recently by Gutman is ex-
tended to the broader concept of generalized thorny graphs. The
latter preserve unchanged certain parts of the parent graph while
applying the procedure of thorn graph generation to the rest of the
graph. Thorn stars and rods are typical examples of such generali-
zed thorn graphs, owing to the unchanged star center, and the other-
than-terminal rod vertices. Formulae are presented for the Wiener
number of these and other classes of (generalized) thorn graphs.
Key words: (generalized) thorn graphs, Wiener number, thorn rings,
thorn stars, thorn rods.
INTRODUCTION
The Wiener number1 is one of the most common topological descriptors
of molecular structure, an area of theoretical chemistry to which Milan Ran-
di} made significant contributions.2,3
Recently, Gutman4,5 introduced the concept of »thorn graph« for any
graph G* that can be obtained from a parent connected graph G by attach-
ing pi  0 new vertices of degree one to each vertex i. Theorems have been
derived for a special case of thorn graphs for which pi = amax – ai, where amax
is a constant, and ai is the degree of the i-th vertex in G, G being a tree. For

Dedicated to Professor Milan Randi} on the occasion of his 70th birthday.
#
Reported in part on the Second Indo-American Workshop on Mathematical Chemistry,
May 30 – June 3, 2000, University of Minnesota, Duluth, Minnesota, USA.
** Author to whom correspondence should be addressed. (E-mail: bonchev@aol.com)
CROATICA CHEMICA ACTA CCACAA 75 (2) 613¿620 (2002)
organic acyclic molecules, amax = 4 or 3 for the respective classes of alkanes
(as in the transformation of 1) or polyenoids (as in 2).
These classes of compounds have only non-terminal vertices of degree t
= amax and terminal vertices of degree 1. Such graphs are sometimes called
proper graphs.6 One might thus regard proper graphs as a special case of
thorny graphs with a uniform vertex degree t of all non-terminal vertices.
We will call such graphs t-thorn graphs. The t-thorn graphs are of impor-
tance for polymer theory, especially for dendrimers.7 The Wiener number of
polymer graphs could be of use in the quantitative characterization of poly-
mer topology,8,9 as well as for specific structure-property relationships. For-
mulae for the Wiener number of dendrimer species have been reported by
Gutman10 and Diudea.11 Recently, Dobrynin, Entringer, and Gutman12 re-
viewed the studies on the Wiener index of trees.
Gutman4 derived a formula that relates the Wiener number W of such
thorn trees T* with the Wiener number of the parent graph T having n ver-
tices:
W(T*) = (t–1)2W(T) + (t–1)n+12 . (1)
Bytautas, Bonchev, and Klein13 calculated the average Wiener number
for the (structural) isomer classes of alkanes and polyenoids having up to
100 carbon atoms, making use of formula (1) and a generation-series-based
methodology for calculating average properties of isomer series.14–15 In this
paper we present closed-form formulae for calculating the Wiener number of
several classes of thorn graphs, such as caterpillars having a chain as a par-
ent graph, and thorn rings whose parent graph is a monocycle. We also ap-
ply the notion of generalized thorn graphs, to consider stars and rods. Such
graphs preserve the vertex degrees in some essential structural fragments
of the parent graph such as the central vertex of a star or the internal part
of a chain, whereas the procedure of the thorn graph formation is applied to
614 D. BONCHEV AND D. J. KLEIN
1
2
the rest of the graph vertices. The Wiener number formulae for these two
classes of general graphs are also presented. The formulae are given in a
form containing essential structural information; besides the number of
thorns these are the rod length and the number of star arms.
THE WIENER NUMBER FORMULAE FOR THORN TREES
Caterpillars
Definition: A caterpillar is a thorn tree T*(a,b) whose nonterminal vertices
b  1 are of the same degree a > 2, and whose parent graph is the b-site
chain graph, Pb.
Theorem: The Wiener number of a caterpillar having b branched vertices of
degree a is:
W(T*(a,b)) =
1
6
b(a–1)(a–1)(b–1)(b+7) + 6(a+1) + 1 . (2)
Proof: We apply the formula originally used by Wiener:
W N Ni j 
1
2 all edges
(3)
where Ni and Nj are the numbers of vertices in the left and right compo-
nents to which the graph is decomposed after cutting a graph edge i, j, and
the sum of Eq. (3) is over all edges in the graph. The total number of verti-
ces in T*(a,b) is
n = (a–1)b + 2 . (4)
Applying the Wiener constructive procedure one obtains
W = 1  (n–1)  (a–2)b+2 + a  (n–a) + (2a–1)  n–(2a–1) +
(3a–2)  n–(3a–2) + ... + 	b–1)a – (b–2)  n – (b–1)a + (b–2) =
(a–1)b+1(a–2)b+2 +
i
b




1
1
i(a–1) + 1(a–1)(b–i) + 1
WIENER NUMBER OF THORN GRAPHS 615
a b= 3, = 6 a b= 4, = 7
wherefrom one arrives after some elementary transformations at formula
(2). For the chemically relevant cases of a = 3 and a = 4 one obtains the Wie-
ner number as a polynomials of degree three in the number of branched ver-
tices:
W(b,a = 3) =
1
3
(2b3 + 12b2 + 10b + 3) (2a)
W(b,a = 4) =
1
2
(3b3 + 18b2 + 9b + 2) . (2b)
It can be shown that our formula (2) along with the formula for W(Pb) is
a consequence of Gutman’s formula (1).
Thorny Rings
Definition: A t-thorny ring has a simple cycle as the parent, and t-2 thorns
at each cycle vertex.
Theorem: The Wiener number of a t-thorny ring having n ring-vertices is
W =
1
8
n(t–1)2 (n2–a) + n(t–2)(nt–n–1) (5)
with a = 0 for n – even, and a = 1 for n – odd.
Proof: Due to symmetry of the t-thorn ring, the Wiener number may be ex-
pressed by the sum d1 of the distances to a ring vertex 1 and that d2 to a ter-
minal vertex 2:
W(Cn,t) =
1
2
nd1 + (t–2)d2 . (6)
By induction, one has
d2 = d1 + (t–1)n–2 (7)
616 D. BONCHEV AND D. J. KLEIN
n t= 3, = 3 n t= 4, = 3 n t= 5, = 3 n t= 6, = 3
d1(n–even) =
1
4
(t–1)n2 + 4(t–2)n;
d1(n–odd) =
1
4
(t–1)n2 + 4(t–2)n – t +1 . (8)
Substituting (7) and (8) into (6) one obtains Eq. (5).
Substituting in (5) n3/8 = Cn-even and (n
3–n)/8 = Cn-odd, one can express
the Wiener number of a t-thorn ring, W(Cn,t), by the Wiener number of the
parent cyclic graph, W(Cn):
W(Cn,t) = (t–1)
2 W(Cn) + n(t–2) (nt – n – 1) . (9)
Eq. (9) for the thorn rings coincides with the equation given as Corollary
1.1 by Gutman4 for the general case of thorn graphs G* obtained from a par-
ent graph G by attaching the same number of vertices t at each of the verti-
ces of G.
Thorn Rods
Definition: A thorn rod is a graph, Pp,t, which includes a linear chain (ter-
med »rod«) of p vertices and degree-t terminal vertices at each of the two rod
ends.
Theorem: The Wiener number of a thorn rod Pp,t is
W(Pp,t) =
1
6
(p3–p) + (t–1)2 (p+3) + (t–1)(p2+p–2) . (10)
Proof: We proceed from the Doyle-Graver formula16
W
n
n n ni j k
i j k
u












 

1
3
. (11)
Here, n is the total number of vertices in the graph, and ni, nj, and nk
are the numbers of vertices in the branches i, j, and k emanating from ver-
tex u, and the first summation is over all branched vertices. The thorn rod
WIENER NUMBER OF THORN GRAPHS 617
p t= 6, = 3 p t= 7, = 4
has only two branched vertices each one having t–1 branches of a single ver-
tex and a large branch with p + t – 2 vertices. Then, from (11) one gets
W(Pp,t) =
n t t
































1
3
2
1
3
2
1
2
(p + t – 2) . (12)
After substituting n = p + 2t – 2 and simplifying Eq. (12) one arrives at
Eq. (10). The specific case with p = 2, corresponds to the »pom-pom« model,
widely used in polymer theory:
W(P2,t) = 5(t–1)
2 + 4(t–1) + 1 . (16a)
Corollary: The Wiener number of the t-thorn rod can be obtained from
the Wiener number of the parent path graph, W(Pp), by the formula:
W(Pp,t) = W(Pp) + (t–1)
2(p+3) + (t–1)(p2+p–2) . (17)
The proof follows directly from Eq. (16) the first term in which is exactly
the Wiener number of the parent path graph.
Thorn Stars
Definition: Thorn stars are graphs obtained from a k-arm star by attaching
t-1 terminal vertices to each of the star arms.
Theorem: The Wiener number of a thorn star Sk,t is
W(Sk,t) = kt(2kt–k–t+1) (18)
Proof: We proceed again from the Doyle-Graver formula (11) for the Wiener
number of a branched tree. There are k+1 branched vertices in Sk,t, k of
which are equivalent and have t–1 branches of one vertex and one branch of
618 D. BONCHEV AND D. J. KLEIN
k t= 3, = 4 k t= 4, = 3
n-t vertices. The central vertex has k equivalent branches, each one with t
vertices. Therefore, (11) transforms into
W =
n
k
t
n t
t




















 

















1
3
1
3
1
2
( ) 









k
t
3
3 . (19)
Expressing the total number of vertices in Sk,t as
n = kt + 1 (20)
and substituting them in (19), one proves the theorem.
Eq. (20) shows that for a given star, the Wiener number is a quadratic
function of the number of thorns t. For the chemically relevant cases of 3-
arm star and 4-arm star one obtains
W(S3,t) = 15(t–1)
2 + 24(t–1) + 9 (18a)
W(S4,t) = 28(t–1)
2 + 44(t–1) + 16 (18b)
Corollary: The Wiener number of the t-thorn star, W(Sk,t), is related to
the Wiener number of the parent star, W(Sk), by the equation
W(Sk,t) = W(Sk) + k(t–1)2k(t–1) + 3k–t (21)
The proof follows directly from comparing Eq. (18) to the equation17
W(Sk) = k
2 . (22)
REFERENCES
1. (a) H. Wiener, J. Am. Chem. Soc. 69 (1947) 17–20; (b) J. Phys. Chem. 52 (1948)
1082–1089.
2. M. Randi}, J. Chem. Inf. Comput. Sci. 33 (1993) 709–716.
3. M. Randi}, J. Chem. Inf. Comput. Sci. 34 (1994) 361–367.
4. I. Gutman, Publ. Inst. Math. (Beograd) 63 (1998) 31–36.
5. I. Gutman, D. Vidovi}, and L. Popovi}, J. Chem. Soc., Faraday Trans. 94 (1998)
857–860.
6. F. Harary, Graph Theory, 2
nd
ed., Addison-Wesley, Reading, MA, 1969.
7. M. V. Diudea and G. Katona, Adv. Dendritic Macromol. 4 (1999) 135–201.
8. D. Bonchev, E. Markel, and A. Dekmezian, Polymer 43 (2002) 203–222.
9. D. Bonchev, E. Markel, and A. Dekmezian, J. Chem. Inf. Comput. Sci. 41 (2001)
1274–1285.
10. I. Gutman, Y. N. Yeh, S. L. Lee, and J. C. Chen, MATCH (Commun. Math. Com-
put. Chem.) 30 (1994) 103–115.
WIENER NUMBER OF THORN GRAPHS 619
11. M. W. Diudea, MATCH (Commun. Math. Comput. Chem.) 35 (1995) 71–83.
12. A. A. Dobrynin, R. Entringer, and I. Gutman, Acta Appl. Math. 66 (2001) 211–249.
13. L. Bytautas, D. Bonchev, and D. J. Klein, MATCH (Commun. Math. Comput.
Chem.), in press.
14. L. Bytautas and D. J. Klein, J. Chem. Inf. Comput. Sci. 38 (1998) 1063–1078.
15. L. Bytautas and D. J. Klein, J. Chem. Inf. Comput. Sci. 40 (2000) 471–481.
16. J. K. Doyle and J. E. Graver, Discrete Math. 17 (1977) 147–154.
17. D. Bonchev and N. Trinajsti}, J. Chem. Phys. 67 (1977) 4517–4533.
SA@ETAK
O Wienerovu broju trnovitih stabala, zvijezda, prstenova i prutova
Danail Bonchev i Douglas J. Klein
Graf-teorijska koncepcija o trnovitim stablima poop}ena je u koncepciju o trno-
vitim grafovima. Ona dopu{ta da izvjesni dijelovi izvornoga grafa ostanu nepromije-
njeni prilikom primjene postupka generiranja trnovitih grafova na ostatak izvornoga
grafa. Trnovite zvijezde i prutovi tipi~ni su primjeri poop}enih trnovitih grafova s
nepromijenjenim sredi{tem zvijezde i nedirnutim unutarnjim ~vorovima pruta. Dane
su formule za izra~unavanje Wienerova broja tih i drugih klasa (poop}enih) trnovitih
grafova.
620 D. BONCHEV AND D. J. KLEIN


On the Wiener Number of Thorn Trees, Stars, Rings, and Rods

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On the Wiener Number of Thorn Trees, Stars, Rings, and Rods

Abstract

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Full text

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