On the average eccentricity, the harmonic index and the largest signless Laplacian eigenvalue of a graph
Publish place: Transactions on Combinatorics، Vol: 6، Issue: 4
Publish Year: 1396
نوع سند: مقاله ژورنالی
زبان: English
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تاریخ نمایه سازی: 29 آبان 1400
Abstract:
The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity ecc\left(G\right) of a graph G is the mean value of eccentricities of all vertices of G. The harmonic index H\left(G\right) of a graph G is defined as the sum of \frac{۲}{d_{i}+d_{j}} over all edges v_{i}v_{j} of G, where d_{i} denotes the degree of a vertex v_{i} in G. In this paper, we determine the unique tree with minimum average eccentricity among the set of trees with given number of pendent vertices and determine the unique tree with maximum average eccentricity among the set of n-vertex trees with two adjacent vertices of maximum degree \Delta, where n\geq ۲\Delta. Also, we give some relations between the average eccentricity, the harmonic index and the largest signless Laplacian eigenvalue, and strengthen a result on the Randi\'{c} index and the largest signless Laplacian eigenvalue conjectured by Hansen and Lucas \cite{hl}.
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Authors
Hanyuan Deng
College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan ۴۱۰۰۸۱, P. R. China
S. Balachandran
Department of Mathematics, School of Humanities and Sciences,SASTRA University, Thanjavur, India
S. K. Ayyaswamy
Department of Mathematics, School of Humanities and Sciences,SASTRA University, Thanjavur, India
Y. B. Venkatakrishnan
Department of Mathematics, School of Humanities and Sciences,SASTRA University, Thanjavur, India
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