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ICM 2006 Short Communications Abstracts Section 14 ICM 2006 Short Communications Abstracts Section 14 Combinatorics ICM 2006 – Short Communications. Abstracts. Section 14 Reciprocal graphs G. Indulal, A. Vijayakumar* Department of Mathematics, Cochin University of Science and Technology, Cochin-682 022, India [email protected] Eigenvalue of a graph is the eigenvalue of its adjacency matrix. A graph G is reciprocal if the reciprocal of each of its eigenvalue is also an eigenvalue 1 P of G. The Wiener index W (G)of a graph G is defined by W (G) = 2 d∈D d r Q and the Multiplicative Wiener index Wπ(G) by Wπ(G) = d where d6=0∈D D is the distance matrix of G.In this paper, some classes of reciprocal graphs ,an upperbound for their energy,a pair of equi energetic reciprocal graphs on every n ≡ 0 mod (12) and n ≡ 0 mod (16), the Wiener indices and Multiplicative Wiener indices of some classes of reciprocal graphs are discussed. References [1] D.M.Cvetkovi´c,Graphs with least eigenvalue -2; a historical survey and recent developments in maximal exceptional graphs. Linear Algebra Appl. 356 (2002) 189-210. [2] J.R.Dias,Properties and relationships of right-hand mirror-plane fragments and their eigenvectors : the concept of complementarity of molecular graphs,Molecular Physics, 1996, Vo l . 88, No . 2, 407- 417 [3] I. Gutman, Topology and stability of conjugated hydrocarbons. The depen- dence of total pi-electron energy on molecular topology, J. Serb. Chem. Soc. 70 (2005) 441-456. [4] I.Gutman,W.Linert,I.Lukovits and Z.Tomovi´c,On the Multiplicative Wiener Index and Its Possible Chemical Applications,Monatshefte f¨urChemie 131,421- 427(2000) [5] G. Indulal, A. Vijayakumar: On a Pair of Equienergetic Graphs,(MATCH)Volume 55 (2006) 83-90. ICM 2006 – Madrid, 22-30 August 2006 1 ICM 2006 – Short Communications. Abstracts. Section 14 Spiral chains: A new proof of the four color theorem I. Cahit Faculty of Engineering, Cyprus International University, Lefkose, North Cyprus [email protected] 2000 Mathematics Subject Classification. 05C Acceptable, but due to extensive usage of a computer rather unpleasant proof of the famous four color map problem of Francis Guthrie were settled eventually by W. Appel and K. Haken in 1976 [1]. Using the same method but shortening the proof twenty years later by another team, namely N. Robertson, D. P. Sanders, P. D. Seymour and R. Thomas would not im- prove considerably the readability of the proof either [2]. Thus it has been widely accepted the need of more elegant and readable proof. There are considerable number of equivalent formulations of the problem but none of them promising for a possible non-computer proofs. On the other hand known proofs are used the concept of Kempe chain and reducibility of the configurations which were a century old ideas [3]. With these in mind we have introduced a new concept which we call “spiral chains” in the max- imal planar graphs and instead of sticking to the reducible configurations which were the main complexity issue in the classical proof, we prefer to show the four colorability of the spiral chains by starting coloring from the inner vertices of a maximal planar graphs [4]. We have shown that for any maximal planar graph as long as spiral chains are being used we do not need the fifth color. Finally by the use of spiral chain coloring we have also given an independent proof of Tait’s conjecture that every (bridgeless) cu- bic planar graph has a three -edge -coloring [5]. Henceforth this paper offers another proof of the four color theorem in two ways which is not based on deep and abstract theories from the other branches of mathematics or using computing power of computers, but rather completely on a new idea in graph theory. References [1] K. Appel and W. Haken, “Every planar map is four colorable”, Contemporary Math., 98 (1989). [2] N. Robertson, D. P. Sanders, P. D. Seymour and R. Thomas, “The four colour theorem”, J. Combin. Theory Ser.,B. 70 (1997), 2-44. [3] A. B. Kempe, “On the geographical problem of the four colours”, American J. of Math., 2(3),193-200, 1879. 2 ICM 2006 – Madrid, 22-30 August 2006 ICM 2006 – Short Communications. Abstracts. Section 14 [4] I. Cahit, “Spiral chains: A new proof of the four color theorem”, arXiv preprint, math. CO/0408247, August 18, 2004. [5] I. Cahit, ”Spiral chains: The proofs of Tait’s and Tutte’s three-edge-coloring conjectures”, arXiv preprint, math. CO/0507127 v1, July 6, 2005. ICM 2006 – Madrid, 22-30 August 2006 3 ICM 2006 – Short Communications. Abstracts. Section 14 Green’s function and Poisson kernel on a path E. Bendito, A. Carmona*, A. M. Encinas, J. M. Gesto MAIII, UPC, Jordi Girona Salgado 1-3, 08034 Barcelona, Spain [email protected] 2000 Mathematics Subject Classification. 39A Green’s function and Poisson kernel of networks can be used to study dif- fusion problems on graphs, such as hitting time, chip-firing and discrete Markov chains. The reason is that these problems are written in terms of boundary value problems with respect to the Laplace operator associated with the network and the Green’s function or the Poisson kernel are the re- solvent kernels. Therefore, Green’s functions and Poisson kernels constitute a powerful tool in dealing with a wide range of combinatorial problems, see [2, 3]. The authors studied in [2] general boundary value problems on arbitrary subsets of a network and in particular they obtained a relation between the Green’s function and the Poisson kernel. In general it is not easy to ob- tain and explicit expression for that functions. However, when the network presents symmetries they can be obtained by hand. Our objective here is to obtain explicit expressions for the Green func- tion and Poisson kernel for the Sturm-Liouville problem associated with the Schr¨odinger operator on the path Pn = {0, . , n + 1}. In this case the explicit construction of such functions is a mechanic process since one has two independent solutions of the homogeneous problem associated with the differences equation. Specifically, they are given through Chebyshev Poly- nomials, since they verified a recurrence relation of the type Pi+2(x) = 2 x Pi+1(x) − Pi(x), i ∈ Z. For instance, if we consider the Neumann regular problem, that is, 2(1+q)z(i)−z(i+1)−z(i) = f(i) , i = 1, . , n, z(0)−z(1) = z(n+1)−z(n) = 0, where q > 0, then the Green function is given by V (1 + q) V (1 + q) , 0 ≤ k ≤ s ≤ n, 1 k−1 n−s Gq(k, s) = 2q Un−1 (1 + q) Vs−1 (1 + q) Vn−k (1 + q) , 1 ≤ s ≤ k ≤ n + 1, where Uk and Vk denote the Second and Third kind Chebyshev polynomials, respectively, [1]. 4 ICM 2006 – Madrid, 22-30 August 2006 ICM 2006 – Short Communications. Abstracts. Section 14 References [1] Mason, J.C. and Handscomb, D.C., Chebyshev Polynomials. Chapman & Hall, CRC Press Company, 2003. [2] Bendito, E., Carmona, A., Encinas, A.M., Solving boundary value problems on networks using equilibrium measures, J. Funct. Anal. 17 (2000), 155–176. [3] Chung, F.R.K., Yau, S.T., Discrete Green functions, J. Comb. Theory 91 (2000), 191–214. ICM 2006 – Madrid, 22-30 August 2006 5 ICM 2006 – Short Communications. Abstracts. Section 14 Independent arcs of acyclic orientations of complete r-partite graphs Gerard Jennhwa Chang*, Chen-Ying Lin‡ and Li-Da Tong‡ Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan; ‡Department of Applied Mathematics, National Sun Yet-sen University, 804, Taiwan [email protected] 2000 Mathematics Subject Classification. 05C Suppose D is an acyclic orientation of a graph G. An arc of D is said to be independent if its reversal results another acyclic orientation. Denote i(D) the number of independent arcs in D and N(G) = {i(D): D is an acyclic orientation of G}. Also, let imin(G) be the minimum of N(G) and imax(G) the maximum. While it is known that imin(G) = |V (G)|−1 for any connected graph G, the present paper determines imax(G) for complete r- partite graphs G. We also determine N(G) for balanced complete r-partite graphs G, and find some complete r-partite graphs G whose N(G) is a set of consecutive integers. As a consequence, we answer a question of West that there exist graphs G whose N(G) are not a set of consecutive integers. References [1] K. L. Collins and K. Tysdal, Dependent edges in Mycielski graphs and 4- colorings of 4-skeletons, J. Graph Theory 46 (2004), 285–296. [2] D. C. Fisher, K. Fraughnaugh L. Langley, and D. B. West, The number of dependent arcs in an acyclic orientation, J. Combin. Theory Ser. B 71 (1997), 73–78. [3] K.-W. Lih, C.-Y. Lin and L.-D. Tong, On an interpolation property of outer- planar graphs, Discrete Applied Math. 154 (2006), 166–172. [4] K. Tysdal, Dependent Edges in Acyclic Orientations of Graphs, Ph.D. Thesis, Wesleyan University, Middletown, CT, May 2001. [5] D. B. West, Acyclic orientations of complete bipartite graphs, Discrete Math. 138 (1995), 393–396. 6 ICM 2006 – Madrid, 22-30 August 2006 ICM 2006 – Short Communications. Abstracts. Section 14 Cops, Robber, and Alarms Nancy E. Clarke*, Emma L. Connon Department of Mathematics and Statistics, Acadia University, Wolfville, Nova Scotia, Canada; Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada [email protected] 2000 Mathematics Subject Classification. 05C75, 05C99, 91A43 The two games considered are mixtures of Searching and Cops and Robber.
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