A note on sum of powers of the Laplacian eigenvalues of bipartite graphs

Let G be a simple graph of order N. The normalized Laplacian Estrada index of G is defined as NEE(G)=?Ni=1 e?i?1, where ?1, ?2,... , ?N are the normalized Laplacian eigenvalues of G. In this paper, we give a tight lower bound for NEE of general graphs. We also calculate NEE for a class of treelike fractals, which contains T fractal and Peano basin fractal as its limiting cases. It is shown that NEE scales linearly with the order of the fractal, in line with a best possible lower bound for connected bipartite graphs.

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On the sum of powers of Laplacian eigenvalues of bipartite graphs

Czechoslovak Mathematical Journal ◽

10.1007/s10587-010-0081-8 ◽

2010 ◽

Vol 60 (4) ◽

pp. 1161-1169 ◽

Cited By ~ 13

Author(s):

Bo Zhou ◽

Aleksandar Ilić

Keyword(s):

Bipartite Graphs ◽

Laplacian Eigenvalues

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On Graphs with Three or Four Distinct Normalized Laplacian Eigenvalues

Algebra Colloquium ◽

10.1142/s1005386719000075 ◽

2019 ◽

Vol 26 (01) ◽

pp. 65-82 ◽

Cited By ~ 3

Author(s):

Xueyi Huang ◽

Qiongxiang Huang

Keyword(s):

Bipartite Graphs ◽

Connected Graphs ◽

Unicyclic Graphs ◽

Laplacian Eigenvalues

We characterize all connected graphs with exactly three distinct normalized Laplacian eigenvalues among which one is equal to 1, and determine all connected bipartite graphs with at least one vertex of degree 1 having exactly four distinct normalized Laplacian eigenvalues. In addition, we find all unicyclic graphs with three or four distinct normalized Laplacian eigenvalues.

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Nordhaus-Gaddum Type Inequalities for Laplacian and Signless Laplacian Eigenvalues

The Electronic Journal of Combinatorics ◽

10.37236/4112 ◽

2014 ◽

Vol 21 (3) ◽

Cited By ~ 2

Author(s):

F. Ashraf ◽

B. Tayfeh-Rezaie

Keyword(s):

Bipartite Graph ◽

Bipartite Graphs ◽

Laplacian Eigenvalue ◽

Laplacian Eigenvalues ◽

Signless Laplacian ◽

Appl Math

Let $G$ be a graph with $n$ vertices. We denote the largest signless Laplacian eigenvalue of $G$ by $q_1(G)$ and Laplacian eigenvalues of $G$ by $\mu_1(G)\ge\cdots\ge\mu_{n-1}(G)\ge\mu_n(G)=0$. It is a conjecture on Laplacian spread of graphs that $\mu_1(G)-\mu_{n-1}(G)\le n-1$ or equivalently $\mu_1(G)+\mu_1(\overline G)\le2n-1$. We prove the conjecture for bipartite graphs. Also we show that for any bipartite graph $G$, $\mu_1(G)\mu_1(\overline G)\le n(n-1)$. Aouchiche and Hansen [Discrete Appl. Math. 2013] conjectured that $q_1(G)+q_1(\overline G)\le3n-4$ and $q_1(G)q_1(\overline G)\le2n(n-2)$. We prove the former and disprove the latter by constructing a family of graphs $H_n$ where $q_1(H_n)q_1(\overline{H_n})$ is about $2.15n^2+O(n)$.

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