Partition energy of a graph
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Let G = (V,E) be a graph. Let V1, V2, . . . , Vk be non-empty disjoint subsets of V such that union equal to V . Then {V1, V2, . . . , Vk} is called partition of vertex set V . Using this partition the graph G can be uniquely represented by a matrix called L-matrix Pk(G), whose entries belong to the set {2, 1, 0,−1} and defined as follows: aij = 8>>< >>: 2 if vi and vj are adjacent within the partition Vi, −1 if vi and vj are non-adjacent within the partition Vi, 1 if vi and vj are adjacent between the partition Vi and Vj for i 6= j, 0 otherwise. The eigenvalues of this matrix are called k-partition eigenvalues of G. The k-partition energy EPk (G) is defined as the sum of the absolute values of kpartition eigenvalues of G. We determine partition energy of some known graphs and also obtain bounds for EPk (G).
Abstract
Let G = (V,E) be a graph. Let V1, V2, . . . , Vk be non-empty disjoint subsets of V such that union equal to V . Then {V1, V2, . . . , Vk} is called partition of vertex set V . Using this partition the graph G can be uniquely represented by a matrix called L-matrix Pk(G), whose entries belong to the set {2, 1, 0,−1} and defined as follows: aij = 8>>< >>: 2 if vi and vj are adjacent within the partition Vi, −1 if vi and vj are non-adjacent within the partition Vi, 1 if vi and vj are adjacent between the partition Vi and Vj for i 6= j, 0 otherwise. The eigenvalues of this matrix are called k-partition eigenvalues of G. The k-partition energy EPk (G) is defined as the sum of the absolute values of kpartition eigenvalues of G. We determine partition energy of some known graphs and also obtain bounds for EPk (G).
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