On the Skew Laplacian Energy of a Digraph 1 INTRODUCTION
On the Skew Laplacian Energy of a Digraph 1 INTRODUCTION
On the Skew Laplacian Energy of a Digraph 1 INTRODUCTION
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1914 C. Adiga and M. Smitha<br />
we obtain<br />
This implies<br />
Therefore<br />
2dH(vn−1) =(2n − 4) − ESLH ≤ (2n − 4) − (2n − 6) = 2.<br />
dH(vn−1) =1.<br />
ESL(H) =(2n − 4) − 2=2n − 6.<br />
By <strong>the</strong> induction hypo<strong>the</strong>sis this means that H is <strong>the</strong> path Pn−1 and vn−1<br />
is end vertex <strong>of</strong> this path. This yields that G must be <strong>the</strong> path Pn. This<br />
completes <strong>the</strong> pro<strong>of</strong>.<br />
Acknowledgement: The first author is thankful to Department <strong>of</strong> Science<br />
and Technology, Government <strong>of</strong> India, New Delhi for <strong>the</strong> financial support<br />
under <strong>the</strong> grant DST/SR/S4/MS:490/07.<br />
References<br />
[1] C. Adiga, R. Balakrishnan and Wasin So, The skew energy <strong>of</strong> a graph<br />
(communicated for publication).<br />
[2] R. Grone, R. Merris, V.Sunder, The <strong>Laplacian</strong> spectrum <strong>of</strong> a graph, SIAM<br />
J.Matrix Anal. Appl. 11(1990), 218-238.<br />
[3] R. Merris, <strong>Laplacian</strong> matrices <strong>of</strong> graphs, A survey, Linear Algebra and its<br />
Appl. 197,198 (1994), 143-176.<br />
[4] Mirjana Lazic Kragujevac, <strong>On</strong> <strong>the</strong> <strong>Laplacian</strong> energy <strong>of</strong> a graph, Czechoslovak<br />
Math.Jour. 56(131) (2006), 1207-1213.<br />
Received: February, 2009
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