Distance spectra of neighbourhood corona of graphs

Articles in Press

Document Type : Original paper

Authors

1 Department of Mathematics, Sree Narayana College, Nattika, Kerala-680566, India

2 St. Joseph’s College (Autonomous) Irinjalakuda, Kerala-680121, Affiliated to University of Calicut, India

3 Department of Mathematics, Maharaja’s College, (Government Autonomous) Ernakulam, Kerala-682011, India

Abstract

The neighbourhood corona $G \star H$ of two graphs $G$ and $H$ is obtained by taking one copy of $G$ and $|V(G)|$ copies of $H$ and making all the neighbours of the $i^{\text{th}}$ vertex of $G$ adjacent with all the vertices in the $i^{\text{th}}$ copy of $H$. In this paper we describe the distance eigenvalues and corresponding eigenvectors of $G \star H$ in terms of the adjacency spectrum of $G$ and $H$ when $G$ is a regular triangle-free graph with diameter 2 and $H$ is regular. Several constructions are proposed using line graphs, iterated line graphs, double graphs, strong double graphs and complement graphs to obtain infinitely many distance non-cospectral pairs of distance equienergetic graphs and non-isomorphic pairs of distance cospectral graphs. Also we obtain the distance Laplacian spectrum of $G \star H$ in terms of the distance Laplacian spectrum of $G$ and Laplacian spectrum of $H$ when $G$ is a transmission regular triangle-free graph with diameter 2. Further we find the distance signless Laplacian spectrum of $G \star H$ in terms of the distance signless Laplacian spectrum of $G$ and signless Laplacian spectrum of $H$ when $G$ is a transmission regular triangle-free graph with diameter 2 and $H$ is regular. We also construct infinitely many non-isomorphic pairs of distance Laplacian cospectral graphs and distance signless Laplacian cospectral graphs.

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References

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Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 18 August 2025

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