On the Vertex Multiplication Graphs
M. Saravanan1, K. M. Kathiresan2
1M. Saravanan, Department of Mathematics, Kalasalingam Academy of Research and Education College, Krishnankoil (Tamil Nadu), India.
2K. M. Kathiresan, Director, Centre for Graph Theory, Ayya Nadar Janaki Ammal College, Sivakasi (Tamil Nadu), India.
Manuscript received on 25 November 2019 | Revised Manuscript received on 19 December 2019 | Manuscript Published on 30 December 2019 | PP: 978-982 | Volume-9 Issue-1S4 December 2019 | Retrieval Number: A12091291S419/19©BEIESP | DOI: 10.35940/ijeat.A1209.1291S419
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© The Authors. Blue Eyes Intelligence Engineering and Sciences Publication (BEIESP). This is an open access article under the CC-BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)

Abstract: For any graph , with vertex set { } and a p-tuble of positive integers , the vertex multiplication graph is defined as the graph with vertex set consists of copies of each , where the copies of and are adjacent in if and only if the corresponding vertices and are adjacent in G . In this paper, we prove that the spectrum of is same as that of spectrum of its quotient graph with additional zero eigenvalues with multiplicity , where . Also we prove that the determinant of is minimum for and maximum for . Also we find distance- i spectrum of thorn graphs, , when G is connected – regular graph with diameter 2.
Keywords: Vertex Multiplication, Eigen Values, Quotient Graph.
Scope of the Article: Cryptography and Applied Mathematics