On the distance spectrum of the Kronecker product of distance regular graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913591382245376 |
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| author | Mondal, Priti Prasanna Atik, Fouzul |
| author_facet | Mondal, Priti Prasanna Atik, Fouzul |
| contents | Consider two simple graphs, G1 and G2, with their respective vertex sets V(G1) and V(G2). The Kronecker product forms a new graph with a vertex set V(G1) X V(G2). In this new graph, two vertices, (x, y) and (u, v), are adjacent if and only if xu is an edge in G1 and yv is an edge in G2. While the adjacency spectrum of this product is known, the distance spectrum remains unexplored. This article determines the distance spectrum of the Kronecker product for a few families of distance regular graphs. We find the exact polynomial, which expresses the distance matrix D as a polynomial of the adjacency matrix, for two distance regular graphs, Johnson and Hamming graphs. Additionally, we present families of distance integral graphs, shedding light on a previously posted open problem given by Indulal and Balakrishnan in (AKCE International Journal of Graphs and Combinatorics, 13(3); 230 to 234, 2016). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_19784 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the distance spectrum of the Kronecker product of distance regular graphs Mondal, Priti Prasanna Atik, Fouzul Spectral Theory Consider two simple graphs, G1 and G2, with their respective vertex sets V(G1) and V(G2). The Kronecker product forms a new graph with a vertex set V(G1) X V(G2). In this new graph, two vertices, (x, y) and (u, v), are adjacent if and only if xu is an edge in G1 and yv is an edge in G2. While the adjacency spectrum of this product is known, the distance spectrum remains unexplored. This article determines the distance spectrum of the Kronecker product for a few families of distance regular graphs. We find the exact polynomial, which expresses the distance matrix D as a polynomial of the adjacency matrix, for two distance regular graphs, Johnson and Hamming graphs. Additionally, we present families of distance integral graphs, shedding light on a previously posted open problem given by Indulal and Balakrishnan in (AKCE International Journal of Graphs and Combinatorics, 13(3); 230 to 234, 2016). |
| title | On the distance spectrum of the Kronecker product of distance regular graphs |
| topic | Spectral Theory |
| url | https://arxiv.org/abs/2411.19784 |