On the spectral radius of weighted unicyclic graphs with a positive weight set

Abstract

The spectrum of weighted graphs are often used to solve the problems in the design of networks and electronic circuits. In this article, we identify the (unique) weighted unicyclic graph with the largest spectral radius among the set of all n-vertex weighted unicyclic graphs with a fixed weight set W _n  = {w ₁, w ₂, … , w _n }, where w ₁ = w ₂ = ··· = w _n  > 0 or, w ₁ ⩾ w ₂ ⩾ ··· ⩾ w _n  > 0 and .

Keywords:

• weighted graph
• weighted unicyclic graph
• spectral radius
• Perron vector

AMS Subject Classifications::

• 05C50
• 15A18

Acknowledgements

The authors would like to express their sincere gratitude to the referee for a very careful reading of this article and for all his or her insightful comments and valuable suggestions, which led to a number of improvements in this article. This work is financially supported by the National Natural Science Foundation of China (Grant No. 11071096) and by self-determined research funds of CCNU09Y01005 and CCNU09Y01018 from the colleges' basic research and operation of MOE.