### AIMS Mathematics

2020, Issue 6: 7214-7233. doi: 10.3934/math.2020461
Research article

# Central vertex join and central edge join of two graphs

• Received: 06 July 2020 Accepted: 30 August 2020 Published: 14 September 2020
• MSC : 05C50

• The central graph $C(G)$ of a graph $G$ is obtained by sub dividing each edge of $G$ exactly once and joining all the nonadjacent vertices in $G$. In this paper, we compute the adjacency, Laplacian and signless Laplacian spectra of central graph of a connected regular graph. Also, we define central vertex join and central edge join of two graphs and calculate their adjacency spectrum, Laplacian spectrum and signless Laplacian spectrum. As an application, some new families of integral graphs and cospectral graphs are constructed. In addition to that the Kirchhoff index and number of spanning trees of the new joins are determined.

Citation: Jahfar T K, Chithra A V. Central vertex join and central edge join of two graphs[J]. AIMS Mathematics, 2020, 5(6): 7214-7233. doi: 10.3934/math.2020461

### Related Papers:

• The central graph $C(G)$ of a graph $G$ is obtained by sub dividing each edge of $G$ exactly once and joining all the nonadjacent vertices in $G$. In this paper, we compute the adjacency, Laplacian and signless Laplacian spectra of central graph of a connected regular graph. Also, we define central vertex join and central edge join of two graphs and calculate their adjacency spectrum, Laplacian spectrum and signless Laplacian spectrum. As an application, some new families of integral graphs and cospectral graphs are constructed. In addition to that the Kirchhoff index and number of spanning trees of the new joins are determined.

 [1] C. Adiga, B. R. Rakshith, K. N. Subba Krishna, Spectra of some new graph operations and some new class of integral graphs, Iranian Journal of Mathematical Sciences and Informatics, 13 (2018), 51-65. [2] D. Cvetkovic, M. Doob, H. Sachs, et al., Spectra of graphs:theory and applications, vol. 10, Academic Press, New York, 1980. [3] D. Cvetkovic, S. Simic, P. Rowlinson, An introduction to the theory of graph spectra, Cambridge University Press, 2009. [4] A. Das and P. Panigrahi, Spectra of R-vertex join and R-edge join of two graphs, Discussiones Mathematicae-General Algebra and Applications, 38 (2018), 19-32. doi: 10.7151/dmgaa.1279 [5] J. Lan and B. Zhou, Spectra of graph operations based on R-graph, Linear and Multilinear Algebra, 63 (2015), 1401-1422. doi: 10.1080/03081087.2014.941292 [6] X. Liu and P. Lu, Spectra of subdivision-vertex and subdivision-edge neighbourhood coronae, Linear Algebra Appl., 438 (2013), 3547-3559. doi: 10.1016/j.laa.2012.12.033 [7] X. Liu and Z. Zhang, Spectra of subdivision-vertex and subdivision-edge joins of graphs, Bullettin of the Malaysian Mathematical Sciences Society, 42 (2019), 15-31. doi: 10.1007/s40840-017-0466-z [8] C. McLeman and E. McNicholas, Spectra of coronae, Linear Algebra Appl., 435 (2011), 998-1007. doi: 10.1016/j.laa.2011.02.007 [9] J. V. Vivin, M. M. Akbar Ali, K. Thilagavathi, On harmonious coloring of central graphs, Advances and applications in discrete mathematics, 2 (2008), 17-33.
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沈阳化工大学材料科学与工程学院 沈阳 110142

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