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Volume : V, Issue : VI, July - 2015

THE COMPLEMENTARY DOMINATING ENERGY OF A GRAPH

K. B. Murthy, Puttaswamy and Ahmed Mohammed Naji

By : Laxmi Book Publication

Abstract :

For a graph G, let D⊆ V (G) be a dominating set of G. If V −D contains a dominating set D1 with respect to D, then D1 is called a complementary (inverse) dominating set of G. The smallest cardinality among all complementary dominating sets of G is called the complementary domination number of G and it is denoted by γc(G). In this paper, we study complementary dominating energy ECD(G) of a graph G. We are compute complementary dominating energies of some standard and well-known families of graphs. Upper and lower bounds for ECD(G) are established.

Keywords :


    Article :


    Cite This Article :

    K. B. Murthy, Puttaswamy and Ahmed Mohammed Naji(2015). THE COMPLEMENTARY DOMINATING ENERGY OF A GRAPH. Indian Streams Research Journal, Vol. V, Issue. VI, http://isrj.org/UploadedData/7378.pdf

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    36. C. Adiga, A. Bayad, I. Gutman, S. A. Srinivas, The minimum covering energy of a graph, Kragujevac Journal Science, 34(2012), 39-56.
    37. A. Graovac, I. Gutman and N. Trinajsti´c, Topological Approach to the Chemistry of Conjugated Molecules, Springer, Berlin 1977.
    38. I. Gutman, O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986.
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    42. R. B. Bapat, S. Pati, Energy of a graph is never an odd integer, Bulletin of Kerala Mathematics Association, 1(2011), 129-132.
    43. F. Harary, Graph Theory, Addison Wesley, Massachusetts, 1969.
    44. C. Adiga, A. Bayad, I. Gutman, S. A. Srinivas, The minimum covering energy of a graph, Kragujevac Journal Science, 34(2012), 39-56.
    45. A. Graovac, I. Gutman and N. Trinajsti´c, Topological Approach to the Chemistry of Conjugated Molecules, Springer, Berlin 1977.
    46. I. Gutman, O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986.
    47. I. Gutman, The energy of a graph, Ber. Math-Statist. Sekt. Forschungsz. Graz, 103(1978), 1-22.
    48. I. Gutman, X. Li, J. Zhang, Graph Energy, (Ed-s: M. Dehmer, F. Em-mert), Streib., Analysis of Complex Networks, From Biology to Linguistics, WileyVCH, Weinheim, (2009), 145-174.
    49. T. T. Chelvam and G. S. Grace Prema, Equality of domination and inverse domination numbers, Ars Combin., 95 (2010) 103-111.
    50. R. B. Bapat, Graphs and Matrices, Hindustan Book Agency, 2011.
    51. T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, Inc., New York 1998.
    52. R. B. Bapat, S. Pati, Energy of a graph is never an odd integer, Bulletin of Kerala Mathematics Association, 1(2011), 129-132.
    53. C. Adiga, A. Bayad, I. Gutman, S. A. Srinivas, The minimum covering energy of a graph, Kragujevac Journal Science, 34(2012), 39-56.
    54. A. Graovac, I. Gutman and N. Trinajsti´c, Topological Approach to the Chemistry of Conjugated Molecules, Springer, Berlin 1977.
    55. I. Gutman, O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986.
    56. I. Gutman, The energy of a graph, Ber. Math-Statist. Sekt. Forschungsz. Graz, 103(1978), 1-22.
    57. I. Gutman, X. Li, J. Zhang, Graph Energy, (Ed-s: M. Dehmer, F. Em-mert), Streib., Analysis of Complex Networks, From Biology to Linguistics, WileyVCH, Weinheim, (2009), 145-174.
    58. T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, Inc., New York 1998.
    59. R. B. Bapat, S. Pati, Energy of a graph is never an odd integer, Bulletin of Kerala Mathematics Association, 1(2011), 129-132.
    60. R. B. Bapat, S. Pati, Energy of a graph is never an odd integer, Bulletin of Kerala Mathematics Association, 1(2011), 129-132.
    61. F. Harary, Graph Theory, Addison Wesley, Massachusetts, 1969.
    62. R. B. Bapat, S. Pati, Energy of a graph is never an odd integer, Bulletin of Kerala Mathematics Association, 1(2011), 129-132.
    63. T. T. Chelvam and G. S. Grace Prema, Equality of domination and inverse domination numbers, Ars Combin., 95 (2010) 103-111.
    64. C. Adiga, A. Bayad, I. Gutman, S. A. Srinivas, The minimum covering energy of a graph, Kragujevac Journal Science, 34(2012), 39-56.
    65. A. Graovac, I. Gutman and N. Trinajsti´c, Topological Approach to the Chemistry of Conjugated Molecules, Springer, Berlin 1977.
    66. I. Gutman, O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986.
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    68. I. Gutman, X. Li, J. Zhang, Graph Energy, (Ed-s: M. Dehmer, F. Em-mert), Streib., Analysis of Complex Networks, From Biology to Linguistics, WileyVCH, Weinheim, (2009), 145-174.
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    72. C. Adiga, A. Bayad, I. Gutman, S. A. Srinivas, The minimum covering energy of a graph, Kragujevac Journal Science, 34(2012), 39-56.
    73. A. Graovac, I. Gutman and N. Trinajsti´c, Topological Approach to the Chemistry of Conjugated Molecules, Springer, Berlin 1977.
    74. I. Gutman, O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986.
    75. I. Gutman, The energy of a graph, Ber. Math-Statist. Sekt. Forschungsz. Graz, 103(1978), 1-22.
    76. I. Gutman, X. Li, J. Zhang, Graph Energy, (Ed-s: M. Dehmer, F. Em-mert), Streib., Analysis of Complex Networks, From Biology to Linguistics, WileyVCH, Weinheim, (2009), 145-174.
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    81. C. Adiga, A. Bayad, I. Gutman, S. A. Srinivas, The minimum covering energy of a graph, Kragujevac Journal Science, 34(2012), 39-56.
    82. A. Graovac, I. Gutman and N. Trinajsti´c, Topological Approach to the Chemistry of Conjugated Molecules, Springer, Berlin 1977.
    83. I. Gutman, O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986.
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    94. C. Adiga, A. Bayad, I. Gutman, S. A. Srinivas, The minimum covering energy of a graph, Kragujevac Journal Science, 34(2012), 39-56.
    95. A. Graovac, I. Gutman and N. Trinajsti´c, Topological Approach to the Chemistry of Conjugated Molecules, Springer, Berlin 1977.
    96. I. Gutman, O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986.
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    102. C. Adiga, A. Bayad, I. Gutman, S. A. Srinivas, The minimum covering energy of a graph, Kragujevac Journal Science, 34(2012), 39-56.
    103. A. Graovac, I. Gutman and N. Trinajsti´c, Topological Approach to the Chemistry of Conjugated Molecules, Springer, Berlin 1977.
    104. I. Gutman, O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986.
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    108. T. T. Chelvam and G. S. Grace Prema, Equality of domination and inverse domination numbers, Ars Combin., 95 (2010) 103-111.
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    111. R. B. Bapat, S. Pati, Energy of a graph is never an odd integer, Bulletin of Kerala Mathematics Association, 1(2011), 129-132.
    112. T. T. Chelvam and G. S. Grace Prema, Equality of domination and inverse domination numbers, Ars Combin., 95 (2010) 103-111.
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    117. A. Graovac, I. Gutman and N. Trinajsti´c, Topological Approach to the Chemistry of Conjugated Molecules, Springer, Berlin 1977.
    118. I. Gutman, O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986.
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    124. T. T. Chelvam and G. S. Grace Prema, Equality of domination and inverse domination numbers, Ars Combin., 95 (2010) 103-111.
    125. T. T. Chelvam and G. S. Grace Prema, Equality of domination and inverse domination numbers, Ars Combin., 95 (2010) 103-111.
    126. F. Harary, Graph Theory, Addison Wesley, Massachusetts, 1969.
    127. F. Harary, Graph Theory, Addison Wesley, Massachusetts, 1969.
    128. T. T. Chelvam and G. S. Grace Prema, Equality of domination and inverse domination numbers, Ars Combin., 95 (2010) 103-111.
    129. C. Adiga, A. Bayad, I. Gutman, S. A. Srinivas, The minimum covering energy of a graph, Kragujevac Journal Science, 34(2012), 39-56.
    130. A. Graovac, I. Gutman and N. Trinajsti´c, Topological Approach to the Chemistry of Conjugated Molecules, Springer, Berlin 1977.
    131. I. Gutman, O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986.
    132. I. Gutman, The energy of a graph, Ber. Math-Statist. Sekt. Forschungsz. Graz, 103(1978), 1-22.
    133. F. Harary, Graph Theory, Addison Wesley, Massachusetts, 1969.
    134. R. B. Bapat, S. Pati, Energy of a graph is never an odd integer, Bulletin of Kerala Mathematics Association, 1(2011), 129-132.
    135. F. Harary, Graph Theory, Addison Wesley, Massachusetts, 1969.
    136. I. Gutman, X. Li, J. Zhang, Graph Energy, (Ed-s: M. Dehmer, F. Em-mert), Streib., Analysis of Complex Networks, From Biology to Linguistics, WileyVCH, Weinheim, (2009), 145-174.
    137. T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, Inc., New York 1998.
    138. R. B. Bapat, S. Pati, Energy of a graph is never an odd integer, Bulletin of Kerala Mathematics Association, 1(2011), 129-132.
    139. C. Adiga, A. Bayad, I. Gutman, S. A. Srinivas, The minimum covering energy of a graph, Kragujevac Journal Science, 34(2012), 39-56.
    140. A. Graovac, I. Gutman and N. Trinajsti´c, Topological Approach to the Chemistry of Conjugated Molecules, Springer, Berlin 1977.
    141. I. Gutman, O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986.
    142. I. Gutman, The energy of a graph, Ber. Math-Statist. Sekt. Forschungsz. Graz, 103(1978), 1-22.
    143. T. T. Chelvam and G. S. Grace Prema, Equality of domination and inverse domination numbers, Ars Combin., 95 (2010) 103-111.
    144. F. Harary, Graph Theory, Addison Wesley, Massachusetts, 1969.
    145. A. Graovac, I. Gutman and N. Trinajsti´c, Topological Approach to the Chemistry of Conjugated Molecules, Springer, Berlin 1977.
    146. I. Gutman, O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986.
    147. I. Gutman, The energy of a graph, Ber. Math-Statist. Sekt. Forschungsz. Graz, 103(1978), 1-22.
    148. I. Gutman, X. Li, J. Zhang, Graph Energy, (Ed-s: M. Dehmer, F. Em-mert), Streib., Analysis of Complex Networks, From Biology to Linguistics, WileyVCH, Weinheim, (2009), 145-174.
    149. T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, Inc., New York 1998.
    150. R. B. Bapat, S. Pati, Energy of a graph is never an odd integer, Bulletin of Kerala Mathematics Association, 1(2011), 129-132.
    151. C. Adiga, A. Bayad, I. Gutman, S. A. Srinivas, The minimum covering energy of a graph, Kragujevac Journal Science, 34(2012), 39-56.
    152. T. T. Chelvam and G. S. Grace Prema, Equality of domination and inverse domination numbers, Ars Combin., 95 (2010) 103-111.
    153. F. Harary, Graph Theory, Addison Wesley, Massachusetts, 1969.
    154. C. Adiga, A. Bayad, I. Gutman, S. A. Srinivas, The minimum covering energy of a graph, Kragujevac Journal Science, 34(2012), 39-56.
    155. A. Graovac, I. Gutman and N. Trinajsti´c, Topological Approach to the Chemistry of Conjugated Molecules, Springer, Berlin 1977.
    156. I. Gutman, O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986.
    157. I. Gutman, The energy of a graph, Ber. Math-Statist. Sekt. Forschungsz. Graz, 103(1978), 1-22.
    158. I. Gutman, X. Li, J. Zhang, Graph Energy, (Ed-s: M. Dehmer, F. Em-mert), Streib., Analysis of Complex Networks, From Biology to Linguistics, WileyVCH, Weinheim, (2009), 145-174.
    159. T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, Inc., New York 1998.
    160. R. B. Bapat, S. Pati, Energy of a graph is never an odd integer, Bulletin of Kerala Mathematics Association, 1(2011), 129-132.
    161. T. T. Chelvam and G. S. Grace Prema, Equality of domination and inverse domination numbers, Ars Combin., 95 (2010) 103-111.
    162. F. Harary, Graph Theory, Addison Wesley, Massachusetts, 1969.
    163. C. Adiga, A. Bayad, I. Gutman, S. A. Srinivas, The minimum covering energy of a graph, Kragujevac Journal Science, 34(2012), 39-56.
    164. A. Graovac, I. Gutman and N. Trinajsti´c, Topological Approach to the Chemistry of Conjugated Molecules, Springer, Berlin 1977.
    165. I. Gutman, O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986.
    166. I. Gutman, The energy of a graph, Ber. Math-Statist. Sekt. Forschungsz. Graz, 103(1978), 1-22.
    167. I. Gutman, X. Li, J. Zhang, Graph Energy, (Ed-s: M. Dehmer, F. Em-mert), Streib., Analysis of Complex Networks, From Biology to Linguistics, WileyVCH, Weinheim, (2009), 145-174.
    168. T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, Inc., New York 1998.
    169. R. B. Bapat, S. Pati, Energy of a graph is never an odd integer, Bulletin of Kerala Mathematics Association, 1(2011), 129-132.
    170. T. T. Chelvam and G. S. Grace Prema, Equality of domination and inverse domination numbers, Ars Combin., 95 (2010) 103-111.
    171. F. Harary, Graph Theory, Addison Wesley, Massachusetts, 1969.
    172. C. Adiga, A. Bayad, I. Gutman, S. A. Srinivas, The minimum covering energy of a graph, Kragujevac Journal Science, 34(2012), 39-56.
    173. A. Graovac, I. Gutman and N. Trinajsti´c, Topological Approach to the Chemistry of Conjugated Molecules, Springer, Berlin 1977.
    174. I. Gutman, O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986.
    175. I. Gutman, The energy of a graph, Ber. Math-Statist. Sekt. Forschungsz. Graz, 103(1978), 1-22.
    176. I. Gutman, X. Li, J. Zhang, Graph Energy, (Ed-s: M. Dehmer, F. Em-mert), Streib., Analysis of Complex Networks, From Biology to Linguistics, WileyVCH, Weinheim, (2009), 145-174.
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    178. R. B. Bapat, S. Pati, Energy of a graph is never an odd integer, Bulletin of Kerala Mathematics Association, 1(2011), 129-132.

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