Complexity to Find Wiener Index of Some Graphs

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1 Complexty to Fnd Wener Index of Some Graphs Kalyan Das Department of Mathematcs Ramnagar College, Purba Mednpur, West Bengal, Inda Abstract. The Wener ndex s one of the oldest graph parameter whch s used to study molecular-graph-based structure. Ths parameter was frst proposed by Harold Wener n 1947 to determnng the bolng pont of paraffn. The Wener ndex of a molecular graph measures the compactness of the underlyng molecule. Ths parameter s wde studed area for molecular chemstry. It s used to study the physo-chemcal propertes of the underlyng organc compounds. The Wener ndex of a connected graph s denoted by W(G) and s defned as, that s W(G) s the sum of dstances between all pars (ordered) of vertces of G. In ths paper, we gve the algorthmc dea to fnd the Wener ndex of some graphs, lke cactus graphs and ntersecton graphs, vz. nterval, crcular-arc, permutaton, trapezod graphs. Keywords: Wener ndex, cactus graphs, ntersecton graphs, nterval graphs, crcular-arc graphs, permutaton graphs, trapezod graphs. AMS Mathematcs Subect Classfcaton (2010): 05C12, 05C85 1. Introducton Molecular descrptor s a fnal result of a logc and mathematcal procedure whch transforms chemcal nformaton encoded wth n a symbolc representaton of a molecule nto a useful number or the result of some standardzed experment. The Wener ndex W(G) s a dstance-based topologcal nvarant s also a molecular descrptor, t much used n the study of the structure-property and the structure-actvty relatonshps of varous classes of bochemcally nterestng compounds ntroduced by Harold Wener n 1947 for predctng bolng ponts ( ) of alkanes based on the formula,where are emprcal constants, and w(3) s called path number. It s defned as the half sum of the dstances between all pars of vertces of G[1,12,14] 2. Algorthms to fnd Wener ndex 2.1. Cactus graphs The class of cactus graph s an mportant subclass of general planar graphs. Let G = ( V, E) be a fnte, connected, undrected smple graph of n vertces m edges, where V s the set of vertces and E s the set of edges. A vertex u s called a cutvertex f removal of u and all edges ncdent on u dsconnect the graph. A connected graph wthout a cutvertex s called a non-separable graph. A block of a graph s a maxmal non-separable subgraph. A cycle s a connected graph (or subgraph) n whch every vertex s of degree two. A block whch s a cycle s 13

2 K.Das called a cyclc block. A cactus graph s a connected graph n whch every block s ether an edge or a cycle. A weghted graph G s a graph n whch every edge s assocates wth a weght. Wthout loss of generalty we assume that all weghts are postve. A weghted cactus graph s a weghted, connected graph n whch every block contanng two vertces s an edge and three or more vertces s a cycle. A path of a graph G s an alternatng sequence of dstnct vertces and edges whch begns and ends wth vertces n G. The length of a path s the sum of the weghts of the edges n the path. a path from vertex u to v s a shortest path f there s no other path from u to v wth lower length. The dstance d ( u, v) between vertces u and v s the length of shortest path between u and v n G. Theorem 1. [13] The shortest dstances from a specfed vertex to all other vertces of a weghted cactus graph can be computed n O (n) tme and the all par shortest dstance of a weghted cactus graph can be computed n O ( n 2 ) tme, where n represents the total number of vertces of the graph. Theorem 2. The Wnner ndex of the cactus graphs can be computed n O( n 2 ) tme, where n represents the total number of vertces of the graph. Defnton of Intersecton graphs A graph G = ( V, E) s called an ntersecton graph for a fnte famly F of a nonempty set f there s a one-to-one correspondence between F and V such that two sets n F have non-empty ntersecton f and only f ther correspondng vertces n V are adacent. We call F an ntersecton model of G. For an ntersecton model F, we use G (F ) to denote the ntersecton graph for F. Dependng on the nature or geometrc confguraton of the sets 2, 2, K dfferent types of ntersecton graphs are generated. The most useful ntersecton graphs are Interval graphs ( S s the set of ntervals on a real lne) Tolerance graphs Crcular-arc graphs ( S s the set of arcs on a crcle) Permutaton graphs ( S s the set of lne segments between two lne segments) Trapezod graphs ( S s the set of trapezums between two lne segments) Dsk graphs ( S s the set of crcles on a plane) Crcle graphs ( S s the set of chords wthn a crcle) Chordal graphs ( S s the set of connected subgraphs of a tree) Strng graphs ( S s the set of curves n a plane) S 1 S

3 Wener Index of a Cycle n the Context of Some Graph Operatons Graphs wth boxcty k ( S s the set of boxes of dmenson k ) Lne graphs ( S s the set of edges of a graph) Interval graphs An undrected graph G = ( V, E) s sad to be an nterval graph f the vertex set V can be put nto one-to-one correspondence wth a set I of ntervals on the real lne such that two vertces are adacent n G f and only f ther correspondng ntervals have nonempty ntersecton. That s, there s a bectve mappng f : V I. The set I s called an nterval representaton of G and G s referred to as the nterval graph of I [19]. A large number of work on ntersecton graphs and cactus graphs have been done n [20-37]. Theorem 3. [7] The tme complexty for fndng the dstances between all par of vertces on nterval graphs s O ( n 2 ). Theorem 4. The Wnner ndex of the nterval graphs can be computed n O( n 2 ) tme, where n represents the total number of vertces of the graph Crcular-arc graphs A graph s a crcular-arc graph f there exsts a famly A of arcs around a crcle and a one-to-one correspondence between vertces of G and arcs n A, such that two dstnct vertces are adacent n G f and only f the correspondng arcs ntersect n A. Such a famly of arcs s called an arc representaton for G. A graph G s a proper crcular-arc (PCA) graph f there exsts an arc representaton for G such that no arc s properly ncluded n another. A graphg s a unt crcular-arc (UCA) graph f there exsts an arc representaton for G such that all the arcs are of the same length. Theorem 5. [18] The all-par shortest paths problem on crcular-arc graph s computed n O ( n 2 ) tme. Theorem 6. The Wnner ndex of the crcular-arc graph s computed n O ( n 2 ) tme Permutaton graphs An undrected graph G = ( V, E) wth vertces V = {1,2, K, n} s called a permutaton graph f there exsts a permutaton π on N = {1,2, K, n} such that for all, N, ( 1 )( π ( ) π 1 ( )) < 0 3

4 K.Das f and only f and are oned by an edge n G [19]. Geometrcally, the ntegers 1,2,K, n are drawn n order on a real lne called as upper lne and π (1), π (2), K, π ( n) on a lne parallel to ths lne called as lower lne such that for each N, s drectly below π (). Next, for each V, a lne segment s drawn from on the lower lne to on the upper lne and t s denoted by l (). Then from defnton t follows that there s an edge (, ) n G f and only f the lne segment l () for ntersects the lne segment l ( ) for. Theorem 7. [17] The all-par shortest paths problem on permutaton graphs n O ( n 2 ) tme. Theorem 8. The Wnner ndex of the permutaton graphs can be computed n O( n 2 ) tme Trapezod graphs A trapezod T s defned by four corner ponts [ a, b, c, d ], where a < b and c < d wth a, b lyng on top lne and d c, lyng on bottom lne of a rectangular channel. An undrected graph G = ( V, E) wth vertex set V { v, v, K, v } and edge set = 1 2 n E { e, e, K, e } s called a trapezod graph f a trapezod representaton can be = 1 2 m obtaned such that each vertex v n V corresponds to a trapezod T and and only f the trapezods T and T correspondng to the vertces v and ( v, v ) E f v ntersect. For smplcty the vertces v, v, 1 2 K, vn are represented respectvely by 1, 2, K, n. Thus the edge (, ) E f and only f T and T ntersect n the trapezod representaton. Theorem 9. The tme complexty to fnd all pars shortest dstances on trapezod graphs s O ( n 2 ). Theorem 10. The tme complexty to compute Wnner ndex of a trapezod grapg s O ( n 2 ). REFERENCES 1. A.Graovac and T.Psansk, On the Wener ndex of a graph, Journal of Mathematcal Chemstry, 8 (1991) R.Balakrshnan and K.Renganathan, A Text Book of Graph Theory, Sprnger-Verlag, New York, R.Balakrshnan, The energy of a graph, Lnear Algebra and ts Applcatons, 387 (2004)

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