# A sharp lower bound of the spectral radius of simple graphs

Abstract: Let G be a simple connected graph with n vertices and let p(G) be its spectral radius. The 2-degree of vertex i is denoted by ti, which is the sum of degrees of the vertices adjacent to i. Let Ni = Σj~i tj and Mi = Σj~i Nj. We find a sharp lower bound of p(G), which only contains two parameter Ni and Mi. Our result extends recent known results.

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...Other papers with lower bounds, namely ∑ v∈V w2(v)(2)/ ∑ v∈V d2 v ≤ λ(2)1 [53], ∑ v∈V w3(v)(2)/ ∑ v∈V w2(v)(2) ≤ λ(2)1 [30], ∑ v∈V w4(v)(2)/ ∑ v∈V w3(v)(2) ≤ λ(2)1 [32], and ∑ v∈V wk+1(v)(2)/ ∑ v∈V wk(v)(2) ≤ λ(2)1 [31] consider the sum of squares of walk numbers, but do not mention the corresponding number of walks of the double length (w4/w2 ≤ λ(2)1, w6/w4 ≤ λ(2)1, w8/w6 ≤ λ(2)1, and w2k+2/w2k ≤ λ(2)1)....

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...In several other papers, the sum of squares of walk numbers was considered to obtain the lower bounds ∑ v∈V w2(v) 2/ ∑ v∈V d 2 v ≤ λ21 [YLT04], ∑ v∈V w3(v) 2/ ∑ v∈V w2(v) 2 ≤ λ21 [HZ05],∑ v∈V w4(v) 2/ ∑ v∈V w3(v) 2 ≤ λ21 [Hu09], and ∑ v∈V wk+1(v) 2/ ∑ v∈V wk(v) 2 ≤ λ21 [HTW07]....

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##### References

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### "A sharp lower bound of the spectral..." refers background in this paper

...By the Perron-Frobenius theorem [1, 2], the spectral radius ρ(G) is simple and there is a unique positive unit eigenvector....

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2,094 citations

### "A sharp lower bound of the spectral..." refers background in this paper

...This corollary follows from Corollary 3 (See [3])....

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...For some recent surveys of the known results about this problem and related topics, we refer the reader to [3, 4, 7] and references therein....

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209 citations

151 citations

### "A sharp lower bound of the spectral..." refers background in this paper

...For some recent surveys of the known results about this problem and related topics, we refer the reader to [3, 4, 7] and references therein....

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