IJPAM: Volume 101, No. 4 (2015)

POWER OF A GRAPH AND BINDING NUMBER

H.B. Walikar$^1$, B.B. Mulla$^2$
$^1$Department of Mathematics
Karnatak University
Dharwad, 580003, Karnataka State, INDIA
$^2$Department of Mathematics
Smt. Indira Gandhi College of Engineering
Sector-16, Koparkhairane, Navi Mumbai, 400709, Maharashtra State, INDIA


Abstract. V.G. Kane (see [#!Kane!#]) asked to study the class of graphs satisfying $bind(G^{k})\geq(bind(G))^{k}$. Here we determine binding number of power of various classes of graphs along with the solution for the parametric equation $bind(G^{k})=(bind(G))^{k}$. We also show that $bind(G^{2})=(bind(G))^{2}$ if and only if $G=H^{+}$, where $H^{+}$ is the graph obtained from $H$ by adjoining a pendant edge to each vertex of $H$, by proving that the diophantine equation $mq^{2}-np^{2}=0$ has no nontrivial integral solution. In Theorem 3.4, we prove that there exists no graph $G$ such that $bind(G^{k})=(bind(G))^{k}$ for $k\geq3$ by seeking integral solution for $mq^{k}-np^{k}=0$.

Received: November 11, 2014

AMS Subject Classification: 05C

Key Words and Phrases: binding number, power, hallian index

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DOI: 10.12732/ijpam.v101i4.5 How to cite this paper?

Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2015
Volume: 101
Issue: 4
Pages: 505 - 511


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