单圈图的匹配与Estrada指数
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摘要
谱图理论是代数图论的一个重要分支,它主要研究图的邻接矩阵和]aplace矩阵的谱性质与结构性质之间的联系,以期通过图的谱参数来刻画图的结构性质.
图的Estrada指数EE(G)定义为其中λ1(G),λ2(G),...,λ。(G)是G邻接矩阵的特征值,是图重要的谱参数Estracla指数源于生物化学领域.2000年,Ernesto Estrada生衡量蛋白质折叠程度时最先引入Estrada指数.随后,由于它在生物化学、信息科学、复杂网络等领域的广泛应用,Estrada指数很快引起了数学工作者的关注.研究者试图通过建立Estrada指数与图的结构参数之间的联系,刻画图的结构性质.众所周知,图的匹配数是图重要的结构参数,在很多方面有广泛的应用.本文讨论了单圈图的匹配数与Estrada指数之间的关系,刻画了含有完美匹配的单圈图中Estrad(?)指数的极大图和极小图,以及具有小匹配数的单圈图中Estrada指数的极大图.
本文的组织结构如下.在第一章,首先介绍了Estrada指数的研究背景和意义；其次介绍相关概念,术语和符号；最后介绍了Estrad(?)指数相关的基本结论,研究进展,以及本文的主要研究结果.第二章讨论了含有完美匹配的单圈图的Estrada指数,给出了含有完美匹配的单圈图中Estrad(?)指数的极大图和极小图；第三章研究了小匹配数的单圈图的Estrad(?)指数,分别刻画了匹配数为2和匹配数为3的单圈图中Estrada指数的极大图.
Spectral graph theory is an important branch of algebraic graph theory which mainly deal with the relation between the structural property and the spectral property of graphs by using the spectrum of adjacency matrix and Laplacian matrix of graphs.
The Estrada index of a graph G is defined as EE(G)=Σin=1eλi(G)which is an important spectral parameter of graphs, where λ1(G).λ2(G),...,λn(G) are the eigenvalues of its adjacency matrix. The Estrada index derived from the field of biochemistry. The Estrada index is an index which was introduced by Ernesto Estrada in2000to measure the degree of folding of a protein. Subsequently, The Estrada index received much attention since it was applied more and more widely in biochemistry, information science, complex network. Investigators try to characterize the structure of property of graphs by creating the relation between the Estrada index and the structural parameter of graphs. As we all know, the matching number of graph is an important structural parameter, which has a wide range of applications in many ways. This dissertation discusses the relationship between the the matching number of unicyclic graphs and Estrada index, characterizes the maximizing graph and minimizing graph among all unicyclic graphs with perfect matching, and the maximizing graph among all unicyclic graph with given small matching number.
The thesis is organized as follows. In Chapter one, we introduce the background of the Estrada index of graphs, some concepts and notations which will be used in following Chap-ter. Finally, we introduce the problems, its development and our main results. In Chapter two, we mainly investigate the Estrada index of unicyclic graph with perfect matching, char-acterize the graphs whose Estrada index attain maximum and minimum among all unicyclic graphs with perfect matching, respectively. In the last Chapter, we discuss the unicyclic graph with given small matching number, characterize the maximizing graphs in unicyclic graph with matching number2and in unicyclic graph with matching number3, respectively.
图的Estrada指数EE(G)定义为其中λ1(G),λ2(G),...,λ。(G)是G邻接矩阵的特征值,是图重要的谱参数Estracla指数源于生物化学领域.2000年,Ernesto Estrada生衡量蛋白质折叠程度时最先引入Estrada指数.随后,由于它在生物化学、信息科学、复杂网络等领域的广泛应用,Estrada指数很快引起了数学工作者的关注.研究者试图通过建立Estrada指数与图的结构参数之间的联系,刻画图的结构性质.众所周知,图的匹配数是图重要的结构参数,在很多方面有广泛的应用.本文讨论了单圈图的匹配数与Estrada指数之间的关系,刻画了含有完美匹配的单圈图中Estrad(?)指数的极大图和极小图,以及具有小匹配数的单圈图中Estrada指数的极大图.
本文的组织结构如下.在第一章,首先介绍了Estrada指数的研究背景和意义；其次介绍相关概念,术语和符号；最后介绍了Estrad(?)指数相关的基本结论,研究进展,以及本文的主要研究结果.第二章讨论了含有完美匹配的单圈图的Estrada指数,给出了含有完美匹配的单圈图中Estrad(?)指数的极大图和极小图；第三章研究了小匹配数的单圈图的Estrad(?)指数,分别刻画了匹配数为2和匹配数为3的单圈图中Estrada指数的极大图.
Spectral graph theory is an important branch of algebraic graph theory which mainly deal with the relation between the structural property and the spectral property of graphs by using the spectrum of adjacency matrix and Laplacian matrix of graphs.
The Estrada index of a graph G is defined as EE(G)=Σin=1eλi(G)which is an important spectral parameter of graphs, where λ1(G).λ2(G),...,λn(G) are the eigenvalues of its adjacency matrix. The Estrada index derived from the field of biochemistry. The Estrada index is an index which was introduced by Ernesto Estrada in2000to measure the degree of folding of a protein. Subsequently, The Estrada index received much attention since it was applied more and more widely in biochemistry, information science, complex network. Investigators try to characterize the structure of property of graphs by creating the relation between the Estrada index and the structural parameter of graphs. As we all know, the matching number of graph is an important structural parameter, which has a wide range of applications in many ways. This dissertation discusses the relationship between the the matching number of unicyclic graphs and Estrada index, characterizes the maximizing graph and minimizing graph among all unicyclic graphs with perfect matching, and the maximizing graph among all unicyclic graph with given small matching number.
The thesis is organized as follows. In Chapter one, we introduce the background of the Estrada index of graphs, some concepts and notations which will be used in following Chap-ter. Finally, we introduce the problems, its development and our main results. In Chapter two, we mainly investigate the Estrada index of unicyclic graph with perfect matching, char-acterize the graphs whose Estrada index attain maximum and minimum among all unicyclic graphs with perfect matching, respectively. In the last Chapter, we discuss the unicyclic graph with given small matching number, characterize the maximizing graphs in unicyclic graph with matching number2and in unicyclic graph with matching number3, respectively.
引文
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[3]R. Carbo-Dorca, Smooth fuction topological structure descriptors based on graph-spectra, J. Math. Chem.44 (2008),373-378.
[4]A. Chang, F. Tian, On the spectral radius of unicyclic graphs with perfect matchings, Linear Algebra Appl.370 (2003),237-250.
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[46]I. Gutman, S. Radenkovic, B. Furtula, T. Mansour and M. Schork, Relating Estrada index with spectral radius, J. Serb. Chem. Soc.72 (2007),1321-1327.
[47]Y. Hou, J. Li, Bounds on the largest eigenvalues of trees with a given size of matching, Linear Algebral Appl.342 (2002),203-217.
[48]A. Ilic, D. Stevanovic, The Estrada index of chemical trees,J. Math. Chem.47 (2010),305-314.
[49]Y. Jiang, A. Tang and R. Hoffmann, Evaluation of moments and their application in Huckel molecular orbital theory, Theoret. Chim. Acta.,66 (1984),183-192.
[50]M. Jungsbluth, B. Burghardt, A. K. Hartmann, Fingerprinting networks:Correlations of local and global network properties, Physica A 381 (2007),444-456.
[51]V. V. Kerrenbroeck, E. Marinari, Ranking vertices or edges of a network by loops:A newap-proach, Phys. Rev. Lett.101 (2008),098701.
[52]J. Li, X. Li, L. Wang, The minimal Estrada index of trees with two maximum degree vertices, MATCH Commun. Math. Comput. Chem.64 (2010),799-810.
[53]J. Li, A note on the maximal Estrada index of trees with a given bipartition, MATCH Commun. Math. Comput. Chem.,66 (2011),765-768.
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[2]H. Bamdad, F. Ashraf and I. Gutman, Lower bounds for Estrada index and Laplacian Estrada index, Applied Mathematics Letters 23 (2010),739-742.
[3]R. Carbo-Dorca, Smooth fuction topological structure descriptors based on graph-spectra, J. Math. Chem.44 (2008),373-378.
[4]A. Chang, F. Tian, On the spectral radius of unicyclic graphs with perfect matchings, Linear Algebra Appl.370 (2003),237-250.
[5]F. Chung and L. Lu, Complex graphs and networks, Conference Board of the Mathematical Sciences Press, U.S.,2006.
[6]D. Cvetkovio, M. Doob, H. Sachs, Spectra of Graphs-Theory and Application, Academic Press, New York,1980; 2nd revised ed:Barth, Heidelberg,1995.
[7]L. Da Fontoura Costa, M.A.R. Tognetti, F.N. Silva, Concentric characterization and clas-sificationof complex network nodes:Application to an institutional collaboration network, Physica.387 (2008),6201-6214.
[8]K. C. Das, S.-G. Lee, On the Estrada index conjecture, Linear Algebra Appl.431 (2009), 1351-1359.
[9]T. DcTslio, Bipartiviy of fullerene graphs and fullerene stability, Chem. Phys. Lett.412 (2005), 336-340.
[10]H. Deng, A proof of a conjecture on Estrada index, MATCH Commun. Math. Comput. Chem. 62 (2009),599-606.
[11]H. Deng, A note on the Estrada indexof trees, MATCH Commun. Math. Comput. Chem.62 (2009),607-610.
[12]H. Deng, S. Radenkovid and I. Gutman, The Estrada index, in:D. Cvetkovic, I. Gutman (Eds.), Applications of Graph Spectra (Math. Inst., Belgrade,2009),123-140.
[13]W. Du, X. Li and Y. Li, The Laplacian energy of random graphs, Journal of Mathematical Analysis and Applications,368 (2010),311-319.
[14]Z. Du, B. Zhou, Minimum Wiener indices of trees and unicyclic graphs of given matching number, Match Commun. Math. Comput. Chem.63 (2010),101-112.
[15]Z. Du, Z. Liu, On the Estrada index and L-Estrada indices of graphs, Linear Algebra Appl. 435 (2011),2065-2076.
[16) Z. Du, B. Zhou, The Estrada index of trees, Linear Algebra Appl.435 (2011),2462-2467.
[17]Z. Du, B. Zhou, On the Estrada index of graphs-with given number of cut edges, Electron. T. Linear Algebra.22 (2011),586-592.
[18]Z. Du, B. Zhou, The Estrada index of unicyclic graphs, Linear Algebra Appl.436 (2011), 3149-3159.
[19]P. Erdos and A. Renyi, On the evolution of random graphs, Publ. Math. Inst. Hungar. Acad. Sci.,5 (1960),17-61.
[20]E. Estrada, Spectral Moments of the Edge Adjacency Matrix in Molecular Graphs.1. Def-inition and Applications to the Prediction of Physical Properties of Alkanes, J, Chem. Inf. Comput. Sci.,36 (1996),844-849.
[21]E. Estrada, Generalized walks-based centrality measures for complex biological networks, Journal of Theoretical Biology 263 (2010),556-565.
[22]E. Estrada, Spectral Moments of the Edge Adjacency Matrix in Molecular Graphs.1. Def-inition and Applications to the Prediction of Physical Properties of Alkanes, J. Chem. Inf. Comput. Sci.36 (1996),844-849.
[23]E. Estrada, Spectral Moments of Edge Adjacency Matrix in Molecular Graphs.2. Molecules Containing Heteroatoms and QSAR Applications. J. Chem. Inf. Comput. Sci.37 (1997), 320-328.
[24]E. Estrada, Spectral Moments of the Edge Adjacency Matrix in Molecular Graphs.3. Molecules Containing Cycles, J. Chem. Inf. Comput. Sci.38 (1998),23-27.
[25]E. Estrada, Characterization of 3D molecular structure, Chem. Phys. Lett.319 (2000),713-718.
[26]E. Estrada, Characterization of the folding degree of proteins, Bioinformatics,18 (2002), 697-704.
[27]E. Estrada, Characterization of the amino acid contribution to the folding degree of proteins, Proteins 54 (2004),727-737.
[28]E. Estrada and J. A. Rodriguez-Velazquez, Subgraph centrality in complex networks, Phys. Rev. E.71 (2005),056103-1-9.
[29]E. Estrada and J. A. Rodriguez-Velazquez, Spectral measures of bipartivity in complex net-works, Phys. Rev. E.72 (2005),046105-1-6.
[30]E. Estrada, J. A. Rodriguez-Velazquez and M. Randic, Atomic branching in molecules, Int. J. Quantum Chem.106 (2006),823-832.
[31]E. Estrada, Topological structural classes of complex networks, Phys. Rev. E 75 (2007),1-12.
[32]E. Estrada and N. Hatano, Statistical-mechanical approach to subgraph centrality in complex networks, Chem. Phys. Lett.439 (2007),247-251.
[33]E. Estrada, Atom-bond connectivity and the energetic of branched alkanes, Chem. Phys. Lett.463 (2008),422-425.
[34]Y.-Z. Fan, Y. Wang, Y.-B. Gao, Minimizing the least eigenvalues of unicyclic graphs with application to spectral spread[J], Linear Algebra Appl.2008,429:577-588.
[35]Y.-Z. Fan, S.-C. Gong, Y. Wang, Y.-B. Gao, First eigenvalue and first eigenvectors of a nonsingular unicyclic mixed graph[J], Discrete Math.2009,309:2479-2487.
[36]G. H. Fath-Tabar, A. R. Ashrafi, I. Gutman, Note on Estrada index and L-Estrada indices of graphs, Bull. cl. Sci. Math.139 (2009),1-16.
[37]G. H. Fath-Tabar, A. R. Ashrafi, New upper bounds for Estrada index of bipartite graphs, Linear Algebra Appl.435 (2011),2607-2611.
[38]G. Ferino, H. Gonzalez-Diaz, G. Delogu, G. Podda and E. Uriarte, Using spectral moments of spiral networks based on PSA/mass spectra outcomes to derive quantitative proteome-disease relationships (QPDRs) and predicting prostate cancer, Biochemical and Biophysical Research Communications,372 (2008),320-325.
[39]D. GomeZ, E. Gonzalez-Aianguena, C. Manuel. Centrality and Power in social network a game theoretic approaeh. Mathematical Social Scienees 2003,46(1):27-54.
[40]A. Gursoy, O. Keskin, R. Nussinov, Topological properties of protein interaction networks from a structural perspective, Biochem. Soc. Trans.36 (2008),1398-1403.
[41]I. Gutman, Lower bounds for Estrada index, Publ. Inst. Math. (Beograd),83 (2008),1-7.
[42]I. Gutman, E. Estrada, J. A. Rodriguez-Velazquez, On a graph-spectrm-based structure descriptor, Croat. Chem. Acta 80 (2007),151-154.
[43]I. Gutman and A. Graovac, Estrada index of cycles and paths, Chem. Phys. Lett.436 (2007), 294-296.
[44]I. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer-Verlag, Berlin,1986.
[45]I. Gutman and S. Radenkovic, A lower bound for the Estrada index of bipartite molecular graphs, Kragujevac J. Sci.29 (2007),67-72.
[46]I. Gutman, S. Radenkovic, B. Furtula, T. Mansour and M. Schork, Relating Estrada index with spectral radius, J. Serb. Chem. Soc.72 (2007),1321-1327.
[47]Y. Hou, J. Li, Bounds on the largest eigenvalues of trees with a given size of matching, Linear Algebral Appl.342 (2002),203-217.
[48]A. Ilic, D. Stevanovic, The Estrada index of chemical trees,J. Math. Chem.47 (2010),305-314.
[49]Y. Jiang, A. Tang and R. Hoffmann, Evaluation of moments and their application in Huckel molecular orbital theory, Theoret. Chim. Acta.,66 (1984),183-192.
[50]M. Jungsbluth, B. Burghardt, A. K. Hartmann, Fingerprinting networks:Correlations of local and global network properties, Physica A 381 (2007),444-456.
[51]V. V. Kerrenbroeck, E. Marinari, Ranking vertices or edges of a network by loops:A newap-proach, Phys. Rev. Lett.101 (2008),098701.
[52]J. Li, X. Li, L. Wang, The minimal Estrada index of trees with two maximum degree vertices, MATCH Commun. Math. Comput. Chem.64 (2010),799-810.
[53]J. Li, A note on the maximal Estrada index of trees with a given bipartition, MATCH Commun. Math. Comput. Chem.,66 (2011),765-768.
[54]C. Y. Lin, C. H. Chin, H. H. Wu, S. H. Chen, CW. Ho, M. T. Ko, HUBBA:Hubs objects analyzer -a framework of interactome hubs identification for network biology, Nucleic Acids Res.36 (2008),438-443.
[55]J. Liu and B. Liu, Bounds of the Estrada index of graphs, Appl. Math. J. Chinese Univ.25 (2010),325-330.
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