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October, 2019 Open Problem on $\sigma$-invariant
Kinkar Ch. Das, Seyed Ahmad Mojallal
Taiwanese J. Math. 23(5): 1041-1059 (October, 2019). DOI: 10.11650/tjm/181104

Abstract

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 $\sigma = 1$.

Citation

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Kinkar Ch. Das. Seyed Ahmad Mojallal. "Open Problem on $\sigma$-invariant." Taiwanese J. Math. 23 (5) 1041 - 1059, October, 2019. https://doi.org/10.11650/tjm/181104

Information

Received: 18 May 2018; Revised: 13 October 2018; Accepted: 11 November 2018; Published: October, 2019
First available in Project Euclid: 21 November 2018

zbMATH: 07126937
MathSciNet: MR4012368
Digital Object Identifier: 10.11650/tjm/181104

Subjects:
Primary: 05C50

Rights: Copyright © 2019 The Mathematical Society of the Republic of China

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Vol.23 • No. 5 • October, 2019
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