CHARACTERISTIC POLYNOMIAL OF SOME CLUSTER GRAPHS
|
|
- Lisa Eaton
- 5 years ago
- Views:
Transcription
1 Kragujevac Journal of Mathematics Volume 37(2) (2013), Pages CHARACTERISTIC POLYNOMIAL OF SOME CLUSTER GRAPHS PRABHAKAR R HAMPIHOLI 1 AND BASAVRAJ S DURGI 2 Abstract The characteristic polynomial of a graph G with p vertices is defined as φ(g : λ) = det(λi A(G)), where A is the adjacency matrix of G and I is the unit matrix The roots of the characteristic equation φ(g : λ) = 0, denoted by λ 1, λ 2,, λ p are the eigenvalues of G The graphs with large number of edges are referred as graph representations of inorganic clusters, called as Cluster graphs In this paper we obtain the characteristic polynomial of class of cluster graphs 1 Introduction Let G be a simple undirected graph with p vertices and q edges Let V (G) = {v 1, v 2,, v p } be the vertex set of G The adjacency matrix of G is defined as A(G) = [a ij ], where a ij = 1 if v i is adjacent to v j and a ij = 0, otherwise The characteristic polynomial of G is φ(g : λ) = det(λi A(G)), where I is an unit matrix of order p The roots of an equation φ(g : λ) = 0 denoted by λ 1, λ 2,, λ p are the eigenvalues of G and their collection is the spectrum of G [1] Characteristic polynomial of some cluster graphs are obtained in [2 5] In this paper we obtain the characteristic polynomial of some more cluster graphs 2 Some edge deleted cluster graphs I Gutman and L Pavlović [2] introduced four classes of graphs obtained from complete graph by deleting edges For the sake of continuity we produce these here Definition 21 [2] Let v be a vertex of the complete graph K p, p 3 and let e i, i = 1, 2,, k, 1 k p 1, be its distinct edges, all being incident to v The graph Ka p (k) or Cl(p, k) is obtained by deleting e i, i = 1, 2,, k from K p In addition Ka p (0) = K p Key words and phrases Spectra, Complete graph, Cluster graphs 2010 Mathematics Subject Classification 05C50 Received: January 13, 2013 Revised: June 21,
2 370 P R HAMPIHOLI AND B S DURGI Definition 22 [2] Let f i, i = 1, 2,, k, 1 k p/2 be independent edges of the complete graph K p, p 3 The graph Kb p is obtained by deleting f i, i = 1, 2,, k from K p The graph Kb p (0) = K p Definition 23 [2] Let V k be a k- element subset of the vertex set of the complete graph K p, 2 k p, p 3 The graph Kc p (k) is obtained by deleting from K p all the edges connecting pairs of vertices from V k In addition Kc p (0) = Kc p (1) = K p Definition 24 [2] Let 3 k p, p 3 The graph Kd p (k) obtained by deleting from K p, the edges belonging to a k- membered cycle Theorem 21 [2] For p 3 and 3 k p 1, it holds that φ(ka p (k) : λ) = (λ + 1) p 3 [λ 3 (p 3)λ 2 (2p k 3)λ + (k 1)(p 1 k)] Theorem 22 [2] For p 3 and 0 k p/2 it holds that φ(kb p (k) : λ) = λ k (λ + 1) p 2k 1 (λ + 2) k 1 [λ 2 (p 3)λ 2(p k 1)] Theorem 23 [2] For p 3 and 0 k p it holds that φ(kc p (k) : λ) = λ k 1 (λ + 1) p k 1 [λ 2 (p k 1)λ k(p k)] Theorem 24 [2] For p 3 and 3 k p it holds that φ(kd p (k) : λ) = (λ + 1) p k 1 [λ 2 (p 4)λ (3p 2k 3)] k 1 ( ( ) ) 2πi λ + 2 cos + 1 k We introduce here another class of graphs obtained from K p and we denote it by Ka p (p, m, k) Two subgraphs G 1 and G 2 of graph G are independent subgraphs if V (G 1 ) V (G 2 ) is an empty set, where V (G) is the vertex set of the graph G Definition 25 [4] Let (K m ) i, i = 1, 2,, k, 1 k p/m, 1 m p, be independent complete subgraphs with m vertices of the complete graph K p, p 3 The graph K ap (m, k) obtained from K p by deleting all the edges of k independent complete subgraphs (K m ) i, i = 1, 2,, k In addition K ap (m, 0) = K ap (0, k) = K ap (0, 0) = K p We note that for k = 1, K ap (p, m, k) = K ap (p, m) Theorem 25 [4] Let p, m, k be positive integers Let mk p Then for p 3, 0 k p/m, 1 m p, the characteristic polynomial of K ap (p, m, k) is φ(k ap (p, m, k) : λ) = λ mk k (λ+1) p mk 1 (λ+m) k 1 [λ 2 (p m 1)λ m(p+k mk 1)]
3 CHARACTERISTIC POLYNOMIAL OF SOME CLUSTER GRAPHS 371 Definition 26 [5] Let K m1 and K m2 be independent complete subgraphs of K p The graph obtained by deleting all the edges of k 1 + k 2 such that m 1 k 1 + m 2 k 2 p independent complete subgraphs K m1 (k 1 copies) and K m2 (k 2 copies) is denoted by K ap (p, (m 1, k 1 ), (m 2, k 2 )) Theorem 26 [5] Let p, m i, k i, i = 1, 2 are positive integers with 2 m ik i p Then for p 3, the characteristic polynomial of K ap (p, (m 1, k 1 ), (m 2, k 2 )) is φ(k ap (p, (m 1, k 1 ), (m 2, k 2 )) : λ) = λ 2 m ik i 2 k i (λ + 1) p 2 m ik i 1 { [ (λ + m 1 p) + k ] ( 2 )} 2(m 2 2 m 1 m 2 ) (λ + 1) + (m 1 1) m i k i p λ + m 2 (λ + m 1 ) k 1 1 (λ + m 2 ) k 2 Now we proceed to prove the main result Definition 27 Let K mi, i = 1, 2,, l are the independent complete subgraphs of K p The graph obtained by deleting all the edges of l k i, such that l m ik i p independent complete subgraphs K m1 (k 1 copies), K m2 (k 2 copies),, K ml (k l copies) is denoted by K ap (p, (m 1, k 1 ), (m 2, k 2 ),, (m l, k l )) We note that for k 1 = k, m 1 = m, k i = 0 for i = 2, 3,, l it holds that K ap (p, (m 1, k 1 ), (m 2, k 2 ),, (m l, k l )) = K ap (p, m, k) Theorem 27 Let p, m i, k i, i = 1, 2,, l are positive integers with l m ik i p Then for p 3, the characteristic polynomial of K ap (p, (m 1, k 1 ), (m 2, k 2 ),, (m l, k l )) is φ(k ap (p, (m 1, k 1 ), (m 2, k 2 ),, (m l, k l )) : λ) = λ l m ik i l k i (λ + 1) p l m ik i 1 {[ ] ( l l )} k i (m 2 i m i m 1 ) (λ + m 1 p) + (λ + 1) + (m 1 1) m i k i p λ + m i (λ + m 1 ) k 1 1 i=2 l (λ + m i ) k i i=2 Proof Without loss of generality we assume that the vertices of K ap (p, (m 1, k 1 ), (m 2, k 2 ),, (m l, k l )) are the vertices of K p and the edges are the edges of K p that are not there in K m1 (k 1 copies), K m2 (k 2 copies),,k ml (k l copies) The characteristic polynomial of K ap (p, (m 1, k 1 ), (m 2, k 2 ),, (m l, k l )) is
4 372 P R HAMPIHOLI AND B S DURGI λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ By elementary calculations on the above determinant, we get the result Here we give some examples in Table 1 p, (m 1, k 1 ), (m 2, k 2 ),, (m l k l ) The characteristic equations of graphs K ap 9,(2,1),(3,1),(4,1) λ 6 (λ 3 26λ 48) 11,(2,2),(3,1),(4,1) λ 7 (λ 4 44λ 2 152λ 144) 12,(2,1),(3,1),(4,1) λ 6 (λ 6 56λ 4 260λ 3 477λ 2 392λ 120) 15,(3,2),(4,1),(5,1) λ 11 (λ 4 83λ 2 402λ 540) 14,(2,1),(3,1),(4,1),(5,1) λ 10 (λ 4 71λ 2 308λ 360) Table 1 The characteristic equations of the graphs denoted by K ap (p, (m 1, k 1 ), (m 2, k 2 ),, (m l, k l )) Acknowledgement: Authors are thankful to the referees for suggestions References [1] D M Cvetković, M Doob and H Sachs, Spectra of Graphs, Academic Press, New York, 1980 [2] I Gutman and L Pavlović, The energy of some graphs with large number of edges, Bull Acad Serbe Sci Arts (Cl Math Natur) 118 (1999), 35 [3] P R Hampiholi, H B Walikar and B S Durgi, Energy of complement of stars, J Indones Math Soc 19(1) (2013), 15 21
5 CHARACTERISTIC POLYNOMIAL OF SOME CLUSTER GRAPHS 373 [4] H B Walikar and H S Ramane, Energy of some cluster graphs, Kragujevac J Sci 23 (2001), [5] P R Hampiholi and B S Durgi, On the spectra and energy of some cluster graphs, Int J Math Sci Engg Appl 7(2) (2013), Department of Master of Computer Applications, Gogte Institute of Technology, Belgaum , India address: prhampi@yahooin 2 Department of Mathematics, KLE College of Engineering and Technology, Belgaum , India address: durgibs@yahoocom
ENERGY OF SOME CLUSTER GRAPHS
51 Kragujevac J. Sci. 23 (2001) 51-62. ENERGY OF SOME CLUSTER GRAPHS H. B. Walikar a and H. S. Ramane b a Karnatak University s Kittur Rani Chennama Post Graduate Centre, Department of Mathematics, Post
More informationLaplacian Polynomial and Laplacian Energy of Some Cluster Graphs
International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 2, Issue 5, May 2014, PP 448-452 ISSN 2347-307X (Print) & ISSN 2347-3142 (Online) wwwarcjournalsorg Laplacian Polynomial
More informationEQUIENERGETIC GRAPHS
5 Kragujevac J. Math. 26 (2004) 5 13. EQUIENERGETIC GRAPHS Harishchandra S. Ramane, 1 Hanumappa B. Walikar, 2 Siddani Bhaskara Rao, 3 B. Devadas Acharya, 4 Prabhakar R. Hampiholi, 1 Sudhir R. Jog, 1 Ivan
More informationSEIDEL ENERGY OF ITERATED LINE GRAPHS OF REGULAR GRAPHS
Kragujevac Journal of Mathematics Volume 39(1) (015), Pages 7 1. SEIDEL ENERGY OF ITERATED LINE GRAPHS OF REGULAR GRAPHS HARISHCHANDRA S. RAMANE 1, IVAN GUTMAN, AND MAHADEVAPPA M. GUNDLOOR 3 Abstract.
More informationLAPLACIAN ENERGY OF UNION AND CARTESIAN PRODUCT AND LAPLACIAN EQUIENERGETIC GRAPHS
Kragujevac Journal of Mathematics Volume 39() (015), Pages 193 05. LAPLACIAN ENERGY OF UNION AND CARTESIAN PRODUCT AND LAPLACIAN EQUIENERGETIC GRAPHS HARISHCHANDRA S. RAMANE 1, GOURAMMA A. GUDODAGI 1,
More informationON EQUIENERGETIC GRAPHS AND MOLECULAR GRAPHS
Kragujevac J. Sci. 29 2007) 73 84. UDC 541.66:547.318 ON EQUIENERGETIC GRAPHS AND MOLECULAR GRAPHS Hanumappa B. Walikar, a Harishchandra S. Ramane, b Ivan Gutman, c Sabeena B. Halkarni a a Department of
More informationNew Results on Equienergetic Graphs of Small Order
International Journal of Computational and Applied Mathematics. ISSN 1819-4966 Volume 12, Number 2 (2017), pp. 595 602 Research India Publications http://www.ripublication.com/ijcam.htm New Results on
More informationResolvent Energy of Graphs
Resolvent Energy of Graphs I.Gutman 1,2, B.Furtula 1, E.Zogić 2, E.Glogić 2 1 Faculty of Science, University of Kragujevac, Kragujevac, Serbia 2 State University of Novi Pazar, Novi Pazar, Serbia May 19,
More informationBulletin T.CXXII de l Académie Serbe des Sciences et des Arts Classe des Sciences mathématiques et naturelles Sciences mathématiques, No 26
Bulletin T.CXXII de l Académie Serbe des Sciences et des Arts - 001 Classe des Sciences mathématiques et naturelles Sciences mathématiques, No 6 SOME SPECTRAL PROPERTIES OF STARLIKE TREES M. LEPOVIĆ, I.
More informationEstrada Index of Graphs
Estrada Index of Graphs Mohammed Kasim 1 Department of Mathematics, University of Kashan, Kashan, Iran Email: kasimmd@kashanu.ac.ir Fumao Zhang, Qiang Wang Department of Mathematics, Xi an University of
More informationLinear Algebra and its Applications
Linear Algebra and its Applications 432 2010 661 669 Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: wwwelseviercom/locate/laa On the characteristic and
More informationON THE WIENER INDEX AND LAPLACIAN COEFFICIENTS OF GRAPHS WITH GIVEN DIAMETER OR RADIUS
MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 63 (2010) 91-100 ISSN 0340-6253 ON THE WIENER INDEX AND LAPLACIAN COEFFICIENTS OF GRAPHS WITH GIVEN DIAMETER
More informationOn the spectral radii of quasi-tree graphs and quasiunicyclic graphs with k pendent vertices
Electronic Journal of Linear Algebra Volume 20 Volume 20 (2010) Article 30 2010 On the spectral radii of quasi-tree graphs and quasiunicyclic graphs with k pendent vertices Xianya Geng Shuchao Li lscmath@mail.ccnu.edu.cn
More informationSome spectral inequalities for triangle-free regular graphs
Filomat 7:8 (13), 1561 1567 DOI 198/FIL138561K Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://wwwpmfniacrs/filomat Some spectral inequalities for triangle-free
More informationSpectral Characterization of Generalized Cocktail-Party Graphs
Journal of Mathematical Research with Applications Nov., 01, Vol. 3, No. 6, pp. 666 67 DOI:10.3770/j.issn:095-651.01.06.005 Http://jmre.dlut.edu.cn Spectral Characterization of Generalized Cocktail-Party
More informationIn this paper, we will investigate oriented bicyclic graphs whose skew-spectral radius does not exceed 2.
3rd International Conference on Multimedia Technology ICMT 2013) Oriented bicyclic graphs whose skew spectral radius does not exceed 2 Jia-Hui Ji Guang-Hui Xu Abstract Let S(Gσ ) be the skew-adjacency
More informationEnergies of Graphs and Matrices
Energies of Graphs and Matrices Duy Nguyen T Parabola Talk October 6, 2010 Summary 1 Definitions Energy of Graph 2 Laplacian Energy Laplacian Matrices Edge Deletion 3 Maximum energy 4 The Integral Formula
More informationarxiv: v1 [math.co] 26 Sep 2017
SPECTRAL RADIUS OF A STAR WITH ONE LONG ARM arxiv:170908871v1 [mathco] 26 Sep 2017 HYUNSHIK SHIN Abstract A tree is said to be starlike if exactly one vertex has degree greater than two In this paper,
More informationLAPLACIAN ENERGY FOR A BALANCED GRAPH
LAPLACIAN ENERGY FOR A BALANCED GRAPH Dr. K. Ameenal Bibi 1, B. Vijayalakshmi 2 1,2 Department of Mathematics, 1,2 D.K.M College for Women (Autonomous), Vellore 632 001, India Abstract In this paper, we
More information38 PETROVIĆ AND MILEKIĆ In this paper we also determine all minimal generalized line graphs with the property 2 (G) > 1. There are exactly 21 such gra
PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE Nouvelle série, tome 68(82) (2000), 37 45 GENERALIZED LINE GRAPHS WITH THE SECOND LARGEST EIGENVALUE AT MOST 1 Miroslav Petrović and Bojana Milekić Communicated
More informationMaximum k-regular induced subgraphs
R u t c o r Research R e p o r t Maximum k-regular induced subgraphs Domingos M. Cardoso a Marcin Kamiński b Vadim Lozin c RRR 3 2006, March 2006 RUTCOR Rutgers Center for Operations Research Rutgers University
More informationBipartite graphs with at most six non-zero eigenvalues
Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 11 (016) 315 35 Bipartite graphs with at most six non-zero eigenvalues
More informationTHE GRAPHS WHOSE PERMANENTAL POLYNOMIALS ARE SYMMETRIC
Discussiones Mathematicae Graph Theory 38 (2018) 233 243 doi:10.7151/dmgt.1986 THE GRAPHS WHOSE PERMANENTAL POLYNOMIALS ARE SYMMETRIC Wei Li Department of Applied Mathematics, School of Science Northwestern
More informationRELATION BETWEEN ENERGY AND LAPLACIAN ENERGY
MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 59 (200) 343-354 ISSN 0340-6253 RELATION BETWEEN ENERGY AND LAPLACIAN ENERGY Ivan Gutman, a NairMariaMaiadeAbreu,
More informationApplied Mathematics Letters
Applied Mathematics Letters (009) 15 130 Contents lists available at ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml Spectral characterizations of sandglass graphs
More informationCOUNTING RELATIONS FOR GENERAL ZAGREB INDICES
Kragujevac Journal of Mathematics Volume 38(1) (2014), Pages 95 103. COUNTING RELATIONS FOR GENERAL ZAGREB INDICES G. BRITTO ANTONY XAVIER 1, E. SURESH 2, AND I. GUTMAN 3 Abstract. The first and second
More informationThe Normalized Laplacian Estrada Index of a Graph
Filomat 28:2 (204), 365 37 DOI 0.2298/FIL402365L Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat The Normalized Laplacian Estrada
More informationOn the normalized Laplacian energy and general Randić index R 1 of graphs
On the normalized Laplacian energy and general Randić index R of graphs Michael Cavers a Shaun Fallat a Steve Kirkland ab3 a Department of Mathematics and Statistics University of Regina Regina SK Canada
More informationNormalized Laplacian spectrum of two new types of join graphs
Journal of Linear and Topological Algebra Vol. 6, No. 1, 217, 1-9 Normalized Laplacian spectrum of two new types of join graphs M. Hakimi-Nezhaad a, M. Ghorbani a a Department of Mathematics, Faculty of
More informationHOSOYA POLYNOMIAL OF THORN TREES, RODS, RINGS, AND STARS
Kragujevac J. Sci. 28 (2006) 47 56. HOSOYA POLYNOMIAL OF THORN TREES, RODS, RINGS, AND STARS Hanumappa B. Walikar, a Harishchandra S. Ramane, b Leela Sindagi, a Shailaja S. Shirakol a and Ivan Gutman c
More informationEnergy, Laplacian Energy and Zagreb Index of Line Graph, Middle Graph and Total Graph
Int. J. Contemp. Math. Sciences, Vol. 5, 21, no. 18, 895-9 Energy, Laplacian Energy and Zagreb Index of Line Graph, Middle Graph and Total Graph Zhongzhu Liu Department of Mathematics, South China Normal
More informationOn marker set distance Laplacian eigenvalues in graphs.
Malaya Journal of Matematik, Vol 6, No 2, 369-374, 2018 https://doiorg/1026637/mjm0602/0011 On marker set distance Laplacian eigenvalues in graphs Medha Itagi Huilgol1 * and S Anuradha2 Abstract In our
More informationarxiv: v1 [math.co] 30 Dec 2015
Resolvent Energy of Unicyclic, Bicyclic and Tricyclic Graphs arxiv:1512.08938v1 [math.co] 30 Dec 2015 Luiz Emilio Allem 1, Juliane Capaverde 1, Vilmar Trevisan 1, Abstract Ivan Gutman 2,3, Emir Zogić 3,
More informationOn the spectral radius of bicyclic graphs with n vertices and diameter d
Linear Algebra and its Applications 4 (007) 9 3 www.elsevier.com/locate/laa On the spectral radius of bicyclic graphs with n vertices and diameter d Shu-Guang Guo Department of Mathematics, Yancheng Teachers
More informationBicyclic digraphs with extremal skew energy
Electronic Journal of Linear Algebra Volume 3 Volume 3 (01) Article 01 Bicyclic digraphs with extremal skew energy Xiaoling Shen Yoaping Hou yphou@hunnu.edu.cn Chongyan Zhang Follow this and additional
More information1.10 Matrix Representation of Graphs
42 Basic Concepts of Graphs 1.10 Matrix Representation of Graphs Definitions: In this section, we introduce two kinds of matrix representations of a graph, that is, the adjacency matrix and incidence matrix
More informationA note on the Laplacian Estrada index of trees 1
MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 63 (2009) 777-782 ISSN 0340-6253 A note on the Laplacian Estrada index of trees 1 Hanyuan Deng College of
More informationZagreb indices of block-edge transformation graphs and their complements
Indonesian Journal of Combinatorics 1 () (017), 64 77 Zagreb indices of block-edge transformation graphs and their complements Bommanahal Basavanagoud a, Shreekant Patil a a Department of Mathematics,
More informationOn spectral radius and energy of complete multipartite graphs
Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn., ISSN 1855-3974 (electronic edn. ARS MATHEMATICA CONTEMPORANEA 9 (2015 109 113 On spectral radius and energy of complete multipartite
More informationFurther Studies on the Sparing Number of Graphs
Further Studies on the Sparing Number of Graphs N K Sudev 1, and K A Germina 1 Department of Mathematics arxiv:1408.3074v1 [math.co] 13 Aug 014 Vidya Academy of Science & Technology Thalakkottukara, Thrissur
More informationEnergy of Graphs. Sivaram K. Narayan Central Michigan University. Presented at CMU on October 10, 2015
Energy of Graphs Sivaram K. Narayan Central Michigan University Presented at CMU on October 10, 2015 1 / 32 Graphs We will consider simple graphs (no loops, no multiple edges). Let V = {v 1, v 2,..., v
More informationBOUNDS ON THE FIBONACCI NUMBER OF A MAXIMAL OUTERPLANAR GRAPH
BOUNDS ON THE FIBONACCI NUMBER OF A MAXIMAL OUTERPLANAR GRAPH Ahmad Fawzi Alameddine Dept. of Math. Sci., King Fahd University of Petroleum and Minerals, Dhahran 11, Saudi Arabia (Submitted August 1) 1.
More informationA Creative Review on Integer Additive Set-Valued Graphs
A Creative Review on Integer Additive Set-Valued Graphs N. K. Sudev arxiv:1407.7208v2 [math.co] 30 Jan 2015 Department of Mathematics Vidya Academy of Science & Technology Thalakkottukara, Thrissur-680501,
More informationOrdering trees by their largest eigenvalues
Linear Algebra and its Applications 370 (003) 75 84 www.elsevier.com/locate/laa Ordering trees by their largest eigenvalues An Chang a,, Qiongxiang Huang b a Department of Mathematics, Fuzhou University,
More informationApplying block intersection polynomials to study graphs and designs
Applying block intersection polynomials to study graphs and designs Leonard Soicher Queen Mary University of London CoCoA15, Colorado State University, Fort Collins, July 2015 Leonard Soicher (QMUL) Block
More informationOn Degree Sum Energy of a Graph
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 9, No. 3, 2016, 30-35 ISSN 1307-553 www.ejpam.com On Degree Sum Energy of a Graph Sunilkumar M. Hosamani 1,, Harishchandra S. Ramane 2 1 Department
More informationSUPER MEAN NUMBER OF A GRAPH
Kragujevac Journal of Mathematics Volume Number (0), Pages 9 0. SUPER MEAN NUMBER OF A GRAPH A. NAGARAJAN, R. VASUKI, AND S. AROCKIARAJ Abstract. Let G be a graph and let f : V (G) {,,..., n} be a function
More informationMaximal Energy Graphs
Advances in Applied Mathematics 6, 47 5 (001) doi:10.1006/aama.000.0705, available online at http://www.idealibrary.com on Maximal Energy Graphs Jack H. Koolen 1 FSPM-Strukturbildungsprozesse, University
More informationLinear Algebra and its Applications
Linear Algebra and its Applications 435 (2011) 1029 1033 Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Subgraphs and the Laplacian
More informationOn the second Laplacian eigenvalues of trees of odd order
Linear Algebra and its Applications 419 2006) 475 485 www.elsevier.com/locate/laa On the second Laplacian eigenvalues of trees of odd order Jia-yu Shao, Li Zhang, Xi-ying Yuan Department of Applied Mathematics,
More informationCharacteristic polynomials of skew-adjacency matrices of oriented graphs
Characteristic polynomials of skew-adjacency matrices of oriented graphs Yaoping Hou Department of Mathematics Hunan Normal University Changsha, Hunan 410081, China yphou@hunnu.edu.cn Tiangang Lei Department
More informationNullity of Hermitian-Adjacency Matrices of Mixed Graphs
Journal of Mathematical Research with Applications Jan., 2018, Vol. 38, No. 1, pp. 23 33 DOI:10.3770/j.issn:2095-2651.2018.01.002 Http://jmre.dlut.edu.cn Nullity of Hermitian-Adjacency Matrices of Mixed
More informationA lower bound for the Laplacian eigenvalues of a graph proof of a conjecture by Guo
A lower bound for the Laplacian eigenvalues of a graph proof of a conjecture by Guo A. E. Brouwer & W. H. Haemers 2008-02-28 Abstract We show that if µ j is the j-th largest Laplacian eigenvalue, and d
More informationExtremal Graphs for Randić Energy
MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 77 (2017) 77-84 ISSN 0340-6253 Extremal Graphs for Randić Energy Kinkar Ch. Das, Shaowei Sun Department
More informationThe spectra of super line multigraphs
The spectra of super line multigraphs Jay Bagga Department of Computer Science Ball State University Muncie, IN jbagga@bsuedu Robert B Ellis Department of Applied Mathematics Illinois Institute of Technology
More informationOn the spectral radius of graphs with cut edges
Linear Algebra and its Applications 389 (2004) 139 145 www.elsevier.com/locate/laa On the spectral radius of graphs with cut edges Huiqing Liu a,meilu b,, Feng Tian a a Institute of Systems Science, Academy
More informationOn the adjacency matrix of a block graph
On the adjacency matrix of a block graph R. B. Bapat Stat-Math Unit Indian Statistical Institute, Delhi 7-SJSS Marg, New Delhi 110 016, India. email: rbb@isid.ac.in Souvik Roy Economics and Planning Unit
More informationJournal of Mathematical Nanoscience. On borderenergetic and L-borderenergetic graphs
Journal of Mathematical Nanoscienese 7 (2) (2017) 71 77 Journal of Mathematical Nanoscience Available Online at: http://jmathnano.sru.ac.ir On borderenergetic and L-borderenergetic graphs Mardjan Hakimi-Nezhaad
More informationInverse and Disjoint Restrained Domination in Graphs
Intern. J. Fuzzy Mathematical Archive Vol. 11, No.1, 2016, 9-15 ISSN: 2320 3242 (P), 2320 3250 (online) Published on 17 August 2016 www.researchmathsci.org International Journal of Inverse and Disjoint
More informationAN EXAMPLE OF USING STAR COMPLEMENTS IN CLASSIFYING STRONGLY REGULAR GRAPHS
Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.yu/filomat Filomat 22:2 (2008), 53 57 AN EXAMPLE OF USING STAR COMPLEMENTS IN CLASSIFYING STRONGLY REGULAR
More informationThe chromatic number and the least eigenvalue of a graph
The chromatic number and the least eigenvalue of a graph Yi-Zheng Fan 1,, Gui-Dong Yu 1,, Yi Wang 1 1 School of Mathematical Sciences Anhui University, Hefei 30039, P. R. China fanyz@ahu.edu.cn (Y.-Z.
More informationTRANSITIVE AND ABSORBENT FILTERS OF LATTICE IMPLICATION ALGEBRAS
J. Appl. Math. & Informatics Vol. 32(2014), No. 3-4, pp. 323-330 http://dx.doi.org/10.14317/jami.2014.323 TRANSITIVE AND ABSORBENT FILTERS OF LATTICE IMPLICATION ALGEBRAS M. SAMBASIVA RAO Abstract. The
More informationGraphs with given diameter maximizing the spectral radius van Dam, Edwin
Tilburg University Graphs with given diameter maximizing the spectral radius van Dam, Edwin Published in: Linear Algebra and its Applications Publication date: 2007 Link to publication Citation for published
More informationA Sharp Upper Bound on Algebraic Connectivity Using Domination Number
A Sharp Upper Bound on Algebraic Connectivity Using Domination Number M Aouchiche a, P Hansen a,b and D Stevanović c,d a GERAD and HEC Montreal, Montreal, Qc, CANADA b LIX, École Polytechnique, Palaiseau,
More informationLower bounds for the Estrada index
Electronic Journal of Linear Algebra Volume 23 Volume 23 (2012) Article 48 2012 Lower bounds for the Estrada index Yilun Shang shylmath@hotmail.com Follow this and additional works at: http://repository.uwyo.edu/ela
More informationCospectrality of graphs
Linear Algebra and its Applications 451 (2014) 169 181 Contents lists available at ScienceDirect Linear Algebra and its Applications www.elsevier.com/locate/laa Cospectrality of graphs Alireza Abdollahi
More informationIndependent Transversals in r-partite Graphs
Independent Transversals in r-partite Graphs Raphael Yuster Department of Mathematics Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv, Israel Abstract Let G(r, n) denote
More informationON GLOBAL DOMINATING-χ-COLORING OF GRAPHS
- TAMKANG JOURNAL OF MATHEMATICS Volume 48, Number 2, 149-157, June 2017 doi:10.5556/j.tkjm.48.2017.2295 This paper is available online at http://journals.math.tku.edu.tw/index.php/tkjm/pages/view/onlinefirst
More informationEven Cycles in Hypergraphs.
Even Cycles in Hypergraphs. Alexandr Kostochka Jacques Verstraëte Abstract A cycle in a hypergraph A is an alternating cyclic sequence A 0, v 0, A 1, v 1,..., A k 1, v k 1, A 0 of distinct edges A i and
More informationD-EQUIENERGETIC SELF-COMPLEMENTARY GRAPHS
123 Kragujevac J. Math. 32 (2009) 123 131. D-EQUIENERGETIC SELF-COMPLEMENTARY GRAPHS Gopalapillai Indulal 1 and Ivan Gutman 2 1 Department of Mathematics, St. Aloysius College, Edathua, Alappuzha 689573,
More informationNew skew Laplacian energy of a simple digraph
New skew Laplacian energy of a simple digraph Qingqiong Cai, Xueliang Li, Jiangli Song arxiv:1304.6465v1 [math.co] 24 Apr 2013 Center for Combinatorics and LPMC-TJKLC Nankai University Tianjin 300071,
More informationA Survey on Energy of Graphs
Annals of Pure and Applied Mathematics Vol. 8, No. 2, 2014, 183-191 ISSN: 2279-087X (P), 2279-0888(online) Published on 17 December 2014 www.researchmathsci.org Annals of A Survey on Energy of Graphs S.Meenakshi
More informationCO PRIME PATH DECOMPOSITION NUMBER OF A GRAPH
An. Şt. Univ. Ovidius Constanţa Vol. 19(1), 011, 3 36 CO PRIME PATH DECOMPOSITION NUMBER OF A GRAPH K. NAGARAJAN and A. NAGARAJAN Abstract A decomposition of a graph G is a collection ψ of edge-disjoint
More informationIntegral trees of odd diameters
Integral trees of odd diameters E. Ghorbani A. Mohammadian B. Tayfeh-Rezaie arxiv:1011.4666v1 [math.co] 21 Nov 2010 School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box
More informationBulletin of the Iranian Mathematical Society
ISSN: 117-6X (Print) ISSN: 1735-8515 (Online) Bulletin of the Iranian Mathematical Society Vol. 4 (14), No. 6, pp. 1491 154. Title: The locating chromatic number of the join of graphs Author(s): A. Behtoei
More informationGraphs with Integer Matching Polynomial Roots
Graphs with Integer Matching Polynomial Roots S. Akbari a, P. Csikvári b, A. Ghafari a, S. Khalashi Ghezelahmad c, M. Nahvi a a Department of Mathematical Sciences, Sharif University of Technology, Tehran,
More informationELA
UNICYCLIC GRAPHS WITH THE STRONG RECIPROCAL EIGENVALUE PROPERTY S. BARIK, M. NATH, S. PATI, AND B. K. SARMA Abstract. AgraphG is bipartite if and only if the negative of each eigenvalue of G is also an
More informationOn the Least Eigenvalue of Graphs with Cut Vertices
Journal of Mathematical Research & Exposition Nov., 010, Vol. 30, No. 6, pp. 951 956 DOI:10.3770/j.issn:1000-341X.010.06.001 Http://jmre.dlut.edu.cn On the Least Eigenvalue of Graphs with Cut Vertices
More informationTHE EXTREMAL FUNCTIONS FOR TRIANGLE-FREE GRAPHS WITH EXCLUDED MINORS 1
THE EXTREMAL FUNCTIONS FOR TRIANGLE-FREE GRAPHS WITH EXCLUDED MINORS 1 Robin Thomas and Youngho Yoo School of Mathematics Georgia Institute of Technology Atlanta, Georgia 0-0160, USA We prove two results:
More informationON SET-INDEXERS OF GRAPHS
Palestine Journal of Mathematics Vol. 3(2) (2014), 273 280 Palestine Polytechnic University-PPU 2014 ON SET-INDEXERS OF GRAPHS Ullas Thomas and Sunil C Mathew Communicated by Ayman Badawi MSC 2010 Classification:
More informationRegular graphs with a small number of distinct eigenvalues
Regular graphs with a small number of distinct eigenvalues Tamara Koledin UNIVERZITET U BEOGRADU ELEKTROTEHNIČKI FAKULTET This is joint work with Zoran Stanić Bipartite regular graphs Bipartite regular
More informationMilovanović Bounds for Seidel Energy of a Graph
Advances in Theoretical and Applied Mathematics. ISSN 0973-4554 Volume 10, Number 1 (2016), pp. 37 44 Research India Publications http://www.ripublication.com/atam.htm Milovanović Bounds for Seidel Energy
More informationTHE MINIMUM MATCHING ENERGY OF BICYCLIC GRAPHS WITH GIVEN GIRTH
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 46, Number 4, 2016 THE MINIMUM MATCHING ENERGY OF BICYCLIC GRAPHS WITH GIVEN GIRTH HONG-HAI LI AND LI ZOU ABSTRACT. The matching energy of a graph was introduced
More informationLinear Algebra and its Applications
Linear Algebra and its Applications xxx (2008) xxx xxx Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: wwwelseviercom/locate/laa Graphs with three distinct
More informationExtremal Topological Indices for Graphs of Given Connectivity
Filomat 29:7 (2015), 1639 1643 DOI 10.2298/FIL1507639T Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Extremal Topological Indices
More informationLinear Algebra and its Applications
Linear Algebra and its Applications 436 (2012) 4043 4051 Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa On the spectral radius
More informationA LOWER BOUND FOR RADIO k-chromatic NUMBER OF AN ARBITRARY GRAPH
Volume 10, Number, Pages 5 56 ISSN 1715-0868 A LOWER BOUND FOR RADIO k-chromatic NUMBER OF AN ARBITRARY GRAPH SRINIVASA RAO KOLA AND PRATIMA PANIGRAHI Abstract Radio k-coloring is a variation of Hale s
More informationarxiv: v2 [math.co] 30 Jun 2013
A polynomial representation and a unique code of a simple undirected graph arxiv:1304.5847v2 [math.co] 30 Jun 2013 Shamik Ghosh, Raibatak Sen Gupta, M. K. Sen Abstract In this note we introduce a representation
More informationOn the Normalized Laplacian Energy(Randić Energy)
On the Normalized Laplacian Energy(Randić Energy) Selçuk University, Konya/Turkey aysedilekmaden@selcuk.edu.tr SGA 2016- Spectral Graph Theory and Applications May 18-20, 2016 Belgrade, SERBIA Outline
More informationThe smallest k-regular h-edge-connected graphs without 1-factors
The smallest k-regular h-edge-connected graphs without 1-factors John Ganci, Douglas B. West March 30, 2003 Dedicated to the memory of Bill Tutte Abstract The minimum order of a k-regular h-edge-connected
More informationGraph fundamentals. Matrices associated with a graph
Graph fundamentals Matrices associated with a graph Drawing a picture of a graph is one way to represent it. Another type of representation is via a matrix. Let G be a graph with V (G) ={v 1,v,...,v n
More informationCLASSIFICATION OF TREES EACH OF WHOSE ASSOCIATED ACYCLIC MATRICES WITH DISTINCT DIAGONAL ENTRIES HAS DISTINCT EIGENVALUES
Bull Korean Math Soc 45 (2008), No 1, pp 95 99 CLASSIFICATION OF TREES EACH OF WHOSE ASSOCIATED ACYCLIC MATRICES WITH DISTINCT DIAGONAL ENTRIES HAS DISTINCT EIGENVALUES In-Jae Kim and Bryan L Shader Reprinted
More informationv iv j E(G) x u, for each v V(G).
Volume 3, pp. 514-5, May 01 A NOTE ON THE LEAST EIGENVALUE OF A GRAPH WITH GIVEN MAXIMUM DEGREE BAO-XUAN ZHU Abstract. This note investigates the least eigenvalues of connected graphs with n vertices and
More informationNon-Recursively Constructible Recursive Families of Graphs
Non-Recursively Constructible Recursive Families of Graphs Colleen Bouey Department of Mathematics Loyola Marymount College Los Angeles, CA 90045, USA cbouey@lion.lmu.edu Aaron Ostrander Dept of Math and
More information4 CONNECTED PROJECTIVE-PLANAR GRAPHS ARE HAMILTONIAN. Robin Thomas* Xingxing Yu**
4 CONNECTED PROJECTIVE-PLANAR GRAPHS ARE HAMILTONIAN Robin Thomas* Xingxing Yu** School of Mathematics Georgia Institute of Technology Atlanta, Georgia 30332, USA May 1991, revised 23 October 1993. Published
More informationOn graphs with largest Laplacian eigenvalue at most 4
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 44 (2009), Pages 163 170 On graphs with largest Laplacian eigenvalue at most 4 G. R. Omidi Department of Mathematical Sciences Isfahan University of Technology
More informationOn the number of spanning trees of K m n ± G graphs
Discrete Mathematics and Theoretical Computer Science DMTCS vol 8, 006, 35 48 On the number of spanning trees of K m n ± G graphs Stavros D Nikolopoulos and Charis Papadopoulos Department of Computer Science,
More informationApplicable Analysis and Discrete Mathematics available online at GRAPHS WITH TWO MAIN AND TWO PLAIN EIGENVALUES
Applicable Analysis and Discrete Mathematics available online at http://pefmath.etf.rs Appl. Anal. Discrete Math. 11 (2017), 244 257. https://doi.org/10.2298/aadm1702244h GRAPHS WITH TWO MAIN AND TWO PLAIN
More informationarxiv: v2 [math.co] 27 Jul 2013
Spectra of the subdivision-vertex and subdivision-edge coronae Pengli Lu and Yufang Miao School of Computer and Communication Lanzhou University of Technology Lanzhou, 730050, Gansu, P.R. China lupengli88@163.com,
More informationJutirekha Dutta and Rajat Kanti Nath. 1. Introduction
MATEMATIČKI VESNIK MATEMATIQKI VESNIK 69, 3 (2017), 226 230 September 2017 research paper originalni nauqni rad FINITE ROUPS WHOSE COMMUTIN RAPHS ARE INTERAL Jutirekha Dutta and Rajat Kanti Nath Abstract.
More information