Abstract
The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph. The subgraph grafting operation on a graph is a kind of subgraph moving between two vertices of the graph. In this paper, we introduce two new subgraph grafting operations on bipartite graph and show how the graph energy changes under these subgraph grafting operations. As the application of these operations, we determine the trees with the third and fourth minimal energies in the set of trees with given order and domination number.
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