On the Harmonic Index and the Signless Laplacian Spectral Radius of Graphs
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Authors: H. DENG, T. VETRíK AND S. BALACHANDRAN
DOI: 10.46793/KgJMat2102.299D
Abstract:
The harmonic index of a conected graph G is defined as H(G) = ∑ uv∈E(G)
, where E(G) is the edge set of G, d(u) and d(v) are the
degrees of vertices u and v, respectively. The spectral radius of a
square matrix M is the maximum among the absolute values of the
eigenvalues of M. Let q(G) be the spectral radius of the signless
Laplacian matrix Q(G) = D(G) + A(G), where D(G) is the diagonal
matrix having degrees of the vertices on the main diagonal and
A(G) is the (0, 1) adjacency matrix of G. The harmonic index of
a graph G and the spectral radius of the matrix Q(G) have been
extensively studied. We investigate the relationship between the
harmonic index of a graph G and the spectral radius of the matrix
Q(G). We prove that for a connected graph G with n vertices, we
have
Keywords:
Harmonic index, spectral radius, eigenvalue, signless Laplacian matrix.
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