Document Type : Research Paper

**Authors**

Department of Mathematics, Kerman Branch, Islamic Azad University, Kerman, Iran

**Abstract**

For two graphs $\mathrm{G}$ and $\mathrm{H}$ with $n$ and $m$ vertices, the corona $\mathrm{G}\circ\mathrm{H}$ of $\mathrm{G}$ and $\mathrm{H}$ is the graph obtained by taking one copy of $\mathrm{G}$ and $n$ copies of $\mathrm{H}$ and then joining the $i^{th}$ vertex of $\mathrm{G}$ to every vertex in the $i^{th}$ copy of $\mathrm{H}$. The neighborhood corona $\mathrm{G}\star\mathrm{H}$ of $\mathrm{G}$ and $\mathrm{H}$ is the graph obtained by taking one copy of $\mathrm{G}$ and $n$ copies of $\mathrm{H}$ and joining every neighbor of the $i^{th}$ vertex of $\mathrm{G}$ to every vertex in the $i^{th}$ copy of $\mathrm{H}$. In this paper, we define four new extensions of corona and neighborhood corona of two graphs $\mathrm{G}$ and $\mathrm{H}$; named the identity-extended corona, identity-extended neighborhood corona, neighborhood extended corona and neighborhood extended neighborhood corona and then determine the spectrum of their adjacency matrix, where $\mathrm{H}$ is a regular graph. As an application, we exhibit infinite families of integral graphs.

**Keywords**

corona of two graphs, Electronic Journal of Graph Theory and Application. Vol. 4 No. 1 (2016), pp. 101-110.

[2] Ch. Adiga, B.R. Rakshith and K.N. Subba Krishna, Spectra of some new graph operations and some new

classes of integral graphs, Iranian Journal of Mathematical Sciences and Informatics. Vol. 13 No. 1 (2018),

pp. 51-65.

[3] K. Balinska, D. Cvetkovic, Z. Radosavljevic, S. Simic, D. Stevanovic, A survey on integral graphs, Univerzitet u Beogradu. Publikacije Elektrotehnickog Fakulteta. Serija Matematika. Vol. 13 (2002), pp. 42-65.

[4] S. Barik, S. Pati and BK. Sarma, The spectrum of the corona of two graphs, SIAM Journal on Discrete

Mathematics. Vol. 21 (2007), pp. 47-56.

[5] S. Barik, G. Sahoo, On the Laplacian spectra of some variants of corona, Linear Algebra and its Applica-

tions. Vol. 512 (2017), pp. 32-47.

[6] L. Barriere, F. Comellas, C. Dalfo and M. A. Fiol, The hierarchical product of graphs, Discrete Applied

Mathematics. Vol. 157 (2009), pp. 36-48.

[7] AE. Brouwer, WH. Haemers, Spectral of graphs, Springer, New York, (2012).

[8] S.-Y. Cui, G.-X. Tian, The spectrum and the signless Laplacian spectrum of coronae, Linear Algebra and

its Applications. Vol. 437 (2012), pp. 1692-1703.

[9] D. B. S. Cvetkovic, M. Doob and H. Sachs, Spectra of graphs- theory and applications, (Third edition),

Johann Ambrosius Barth, Heidelberg, (1995).

[10] D. B. S. Cvetkovic, P. Rowlinson and H. Simic, An introduction to the theory of graph spectra, Cambridge

University Press, Cambridge, (2010).

[11] W. L. Ferrar, A text-book of determinants, matrices and algebraic forms, (Second edition), Oxford Univer-

sity Press, (1957).

November 2018

Pages 23-34