The minimum vertex-block dominating energy of the graph
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Let B(G) denote the set of all blocks of a graph G. A vertex v 2 V and a block b 2 B(G) are said to block dominate (b-dominate) each other if v is in the block b. A set D 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 vb = vb(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 ( vb-set) of G. Upper and lower bounds for Evb(G) are established and we also obtain energy of some family of graphs.
Abstract
Let B(G) denote the set of all blocks of a graph G. A vertex v 2 V and a block b 2 B(G) are said to block dominate (b-dominate) each other if v is in the block b. A set D 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 vb = vb(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 ( vb-set) of G. Upper and lower bounds for Evb(G) are established and we also obtain energy of some family of graphs.
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