### Abstract

Let B(G) denote the set of all blocks of a graph G. A vertex v G V and a block b G B(G) are said to block dominate (b-dominate) each other if v is in the block b. A set D C V is said to be a vertex block dominating set (VBD-set) if every block in G is b-dominated by some vertex in D. The vertex block domination number 7^5 = b(G) is the cardinality of the minimum vertex block dominating set of G. In this paper we introduce new kind of graph energy, the minimum vertex block dominating energy of the graph denoting it as E_{vb}(G). It depends both on the underlying graph of G and the particular minimum vertex block dominating set (7^-set) of G. Upper and lower bounds for E_{vb}(G) are established and we also obtain energy of some family of graphs.

Original language | English |
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Pages (from-to) | 593-608 |

Number of pages | 16 |

Journal | Proceedings of the Jangjeon Mathematical Society |

Volume | 22 |

Issue number | 4 |

DOIs | |

Publication status | Published - 01-01-2019 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Proceedings of the Jangjeon Mathematical Society*,

*22*(4), 593-608. https://doi.org/10.17777/pjms2019.22.4.593

}

*Proceedings of the Jangjeon Mathematical Society*, vol. 22, no. 4, pp. 593-608. https://doi.org/10.17777/pjms2019.22.4.593

**The minimum vertex-block dominating energy of the graph.** / Udupa, Sayinath; Bhat, R. S.; Madhusudanan, Vinay.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The minimum vertex-block dominating energy of the graph

AU - Udupa, Sayinath

AU - Bhat, R. S.

AU - Madhusudanan, Vinay

PY - 2019/1/1

Y1 - 2019/1/1

N2 - Let B(G) denote the set of all blocks of a graph G. A vertex v G V and a block b G B(G) are said to block dominate (b-dominate) each other if v is in the block b. A set D C V is said to be a vertex block dominating set (VBD-set) if every block in G is b-dominated by some vertex in D. The vertex block domination number 7^5 = b(G) is the cardinality of the minimum vertex block dominating set of G. In this paper we introduce new kind of graph energy, the minimum vertex block dominating energy of the graph denoting it as Evb(G). It depends both on the underlying graph of G and the particular minimum vertex block dominating set (7^-set) of G. Upper and lower bounds for Evb(G) are established and we also obtain energy of some family of graphs.

AB - Let B(G) denote the set of all blocks of a graph G. A vertex v G V and a block b G B(G) are said to block dominate (b-dominate) each other if v is in the block b. A set D C V is said to be a vertex block dominating set (VBD-set) if every block in G is b-dominated by some vertex in D. The vertex block domination number 7^5 = b(G) is the cardinality of the minimum vertex block dominating set of G. In this paper we introduce new kind of graph energy, the minimum vertex block dominating energy of the graph denoting it as Evb(G). It depends both on the underlying graph of G and the particular minimum vertex block dominating set (7^-set) of G. Upper and lower bounds for Evb(G) are established and we also obtain energy of some family of graphs.

UR - http://www.scopus.com/inward/record.url?scp=85075176531&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85075176531&partnerID=8YFLogxK

U2 - 10.17777/pjms2019.22.4.593

DO - 10.17777/pjms2019.22.4.593

M3 - Article

AN - SCOPUS:85075176531

VL - 22

SP - 593

EP - 608

JO - Proceedings of the Jangjeon Mathematical Society

JF - Proceedings of the Jangjeon Mathematical Society

SN - 1598-7264

IS - 4

ER -