Taiwanese Journal of Mathematics

Open Problem on $\sigma$-invariant

Kinkar Ch. Das and Seyed Ahmad Mojallal

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Let $G$ be a graph of order $n$ with $m$ edges. Also let $\mu_1 \geq \mu_2 \geq \cdots \geq \mu_{n-1} \geq \mu_n = 0$ be the Laplacian eigenvalues of graph $G$ and let $\sigma = \sigma(G)$ ($1 \leq \sigma \leq n$) be the largest positive integer such that $\mu_{\sigma} \geq 2m/n$. In this paper, we prove that $\mu_2(G) \geq 2m/n$ for almost all graphs. Moreover, we characterize the extremal graphs for any graphs. Finally, we provide the answer to Problem 3 in [8], that is, the characterization of all graphs with $\sigma = 1$.

Article information

Taiwanese J. Math., Volume 23, Number 5 (2019), 1041-1059.

Received: 18 May 2018
Revised: 13 October 2018
Accepted: 11 November 2018
First available in Project Euclid: 21 November 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)

graph Laplacian matrix second largest Laplacian eigenvalue average degree Laplacian energy $\sigma$-invariant


Das, Kinkar Ch.; Mojallal, Seyed Ahmad. Open Problem on $\sigma$-invariant. Taiwanese J. Math. 23 (2019), no. 5, 1041--1059. doi:10.11650/tjm/181104. https://projecteuclid.org/euclid.twjm/1542790915

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