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--></p><p class="MsoNormal" style="margin-top: 0cm; margin-right: 15.55pt; margin-bottom: .0001pt; margin-left: 14.2pt; text-align: justify; line-height: normal; tab-stops: 347.3pt;"><span style="font-size: 9.0pt; font-family: "Times New Roman","serif";">In a graph G, the distance d(<em style="mso-bidi-font-style: normal;">u,v</em>) between a pair of vertices <em style="mso-bidi-font-style: normal;">u</em> and <em style="mso-bidi-font-style: normal;">v</em> is the length of a shortest path joining them. A vertex <em style="mso-bidi-font-style: normal;">v</em> is a boundary vertex of a vertex <em style="mso-bidi-font-style: normal;">u</em> if <span style="position: relative; top: 5.0pt; mso-text-raise: -5.0pt;"> </span><span style="mso-spacerun: yes;"> </span>for all <span style="position: relative; top: 5.0pt; mso-text-raise: -5.0pt;"> </span><span style="mso-spacerun: yes;"> </span>The boundary graph <em style="mso-bidi-font-style: normal;">B</em>(<em style="mso-bidi-font-style: normal;">G</em>) based on a connected graph <em style="mso-bidi-font-style: normal;">G</em> is a simple graph which has the vertex set as in <em style="mso-bidi-font-style: normal;">G.</em> Two vertices <em style="mso-bidi-font-style: normal;">u</em> and <em style="mso-bidi-font-style: normal;">v </em>are adjacent in B(G)<em style="mso-bidi-font-style: normal;"> </em>if either<em style="mso-bidi-font-style: normal;"> u</em> is a boundary of <em style="mso-bidi-font-style: normal;">v</em> or <em style="mso-bidi-font-style: normal;">v</em> is a boundary of <em style="mso-bidi-font-style: normal;">u.</em> If <em style="mso-bidi-font-style: normal;">G </em>is disconnected, then each vertex in a component is adjacent to all other vertices in the other components and is adjacent to all of its boundary vertices within the component. Given a positive integer <em style="mso-bidi-font-style: normal;">m</em>, the <em style="mso-bidi-font-style: normal;">m<sup>th</sup></em> iterated boundary graph of <em style="mso-bidi-font-style: normal;">G</em> is defined as <span style="position: relative; top: 5.0pt; mso-text-raise: -5.0pt;"> </span><span style="mso-spacerun: yes;"> </span>A graph <em style="mso-bidi-font-style: normal;">G</em> is periodic if <span style="position: relative; top: 5.0pt; mso-text-raise: -5.0pt;"> </span><span style="position: relative; top: 5.0pt; mso-text-raise: -5.0pt;"><span style="mso-spacerun: yes;"> </span></span>for some m. A graph <em style="mso-bidi-font-style: normal;">G</em> is said to be an eventually periodic graph if there exist positive integers <em style="mso-bidi-font-style: normal;">m</em> and <em style="mso-bidi-font-style: normal;">k >0</em> such that <span style="position: relative; top: 5.0pt; mso-text-raise: -5.0pt;"> </span><span style="position: relative; top: 5.0pt; mso-text-raise: -5.0pt;"><span style="mso-spacerun: yes;"> </span></span>We give the necessary and sufficient condition for a graph to be eventually periodic.</span></p> <p class="MsoNormal" style="margin-top: 0cm; margin-right: 15.55pt; margin-bottom: .0001pt; margin-left: 14.2pt; line-height: normal; tab-stops: 347.3pt;"><em style="mso-bidi-font-style: normal;"><span style="font-size: 6.0pt; font-family: "Times New Roman","serif";"> </span></em></p> <p class="MsoNormal" style="margin-top: 0cm; margin-right: 15.55pt; margin-bottom: .0001pt; margin-left: 14.2pt; line-height: normal; tab-stops: 347.3pt;"> </p><p class="MsoNormal" style="margin-top: 0cm; margin-right: 15.55pt; margin-bottom: .0001pt; margin-left: 14.2pt; line-height: normal; tab-stops: 347.3pt;"><em style="mso-bidi-font-style: normal;"><span style="font-size: 9.0pt; font-family: "Times New Roman","serif";">Keyword</span></em><span style="font-size: 9.0pt; font-family: "Times New Roman","serif";">s<strong style="mso-bidi-font-weight: normal;">:<span style="mso-spacerun: yes;"> </span></strong>Boundary graph; Periodic graph.</span></p><p class="MsoNormal" style="margin-top: 0cm; margin-right: 15.55pt; margin-bottom: .0001pt; margin-left: 14.2pt; line-height: normal; tab-stops: 347.3pt;"><span style="font-size: 9.0pt; font-family: "Times New Roman","serif";"><br /></span></p> <p class="MsoNormal" style="margin-top: 0cm; margin-right: 15.55pt; margin-bottom: .0001pt; margin-left: 14.2pt; text-align: justify; line-height: normal; tab-stops: 347.3pt;"><span style="font-size: 5.0pt; mso-bidi-font-size: 10.0pt; font-family: "Times New Roman","serif";"> </span></p> <p class="MsoNormal" style="margin-top: 0cm; margin-right: 0cm; margin-bottom: .0001pt; margin-left: 14.2pt; line-height: normal; mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 8.0pt; font-family: "Times New Roman","serif"; mso-ansi-language: EN-SG; mso-fareast-language: EN-SG;" lang="EN-SG">© 2013 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.</span></p><p class="MsoNormal" style="margin-top: 0cm; margin-right: 0cm; margin-bottom: .0001pt; margin-left: 14.2pt; line-height: normal; mso-layout-grid-align: none; text-autospace: none;"><span style="font-size: 8.0pt; font-family: "Times New Roman","serif"; mso-ansi-language: EN-SG; mso-fareast-language: EN-SG;" lang="EN-SG"><br /></span></p> <p class="MsoNormal" style="margin-top: 0cm; margin-right: 0cm; margin-bottom: .0001pt; margin-left: 14.2pt; line-height: normal; tab-stops: 14.2pt 6.0cm 177.2pt 184.3pt;"><span style="font-size: 1.0pt; mso-bidi-font-size: 7.0pt; font-family: "Times New Roman","serif"; mso-ansi-language: EN-SG; mso-fareast-language: EN-SG;" lang="EN-SG"> </span></p> <p class="MsoNormal" style="margin-top: 0cm; margin-right: 15.55pt; margin-bottom: .0001pt; margin-left: 14.2pt; text-align: justify; line-height: normal; tab-stops: 347.3pt;"><span style="font-size: 8.0pt; font-family: "Times New Roman","serif";">doi: <a href="http://dx.doi.org/10.3329/jsr.v5i3.14866">http://dx.doi.org/10.3329/jsr.v5i3.14866</a><span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span>J. Sci. Res. <strong style="mso-bidi-font-weight: normal;">5</strong> (3), xxx-xxx (2013)</span></p><p> </p>}, number={3}, journal={Journal of Scientific Research}, author={Mariumuthu, G. and Saraswathy, M. S.}, year={2013}, month={Aug.}, pages={447–455} }