General Randic matrix and general Randic energy

Document Type : Research Paper

Authors

1 Center for Combinatorics, nankai University, Tianjin, China

2 Center for Combinatorics, Nankai University, Tianjin, China

3 Center for Combinatorics, Nankai University, Tianjin 300071, China

Abstract

Let $G$ be a simple graph with vertex set $V(G) = \{v_1‎, ‎v_2,\ldots‎, ‎v_n\}$ and $d_i$ the degree of its vertex $v_i$‎, ‎$i = 1‎, ‎2‎, ‎\dots‎, ‎n$‎. ‎Inspired by the Randic matrix and the general Randic‎ ‎index of a graph‎, ‎we introduce the concept of general Randi'c‎ ‎matrix $\textbf{R}_\alpha$ of $G$‎, ‎which is defined by‎ $(\textbf{R}_\alpha)_{i,j}=(d_id_j)^\alpha$ if $v_i$ and $v_j$ are‎ ‎adjacent‎, ‎and zero otherwise‎. ‎Similarly‎, ‎the general Randic‎ ‎eigenvalues are the eigenvalues of the general Randic} matrix‎, ‎the greatest general Randic eigenvalue is the general Randic‎ ‎spectral radius of $G$‎, ‎and the general Randic energy is the sum‎ ‎of the absolute values of the general Randic eigenvalues‎. ‎In ‎this paper‎, ‎we prove some properties of the general Randi'c matrix‎ ‎and obtain lower and upper bounds for general Randic energy‎, ‎also‎, ‎we get some lower bounds for general Randic spectral‎ ‎radius of a connected graph‎. ‎Moreover‎, ‎we give a new sharp upper‎ ‎bound for the general Randic energy when $\alpha=-1/2$‎.

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References

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Transactions on Combinatorics

Volume 3, Issue 3 - Serial Number 3
September 2014
Pages 21-33

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