Lexicographic polynomials of graphs and their spectra
For a (simple) graph $H$ and non-negative integers $c_0,c_1,\ldots,c_d$ ($c_d \neq 0$), $p(H)=\sum_{k=0}^d{c_k \cdot H^k}$ is the lexicographic polynomial in $H$ of degree $d$, where the sum of two graphs is their join and $c_k \cdot H^k$ is the join of $c_k$ copies of $H^k$. The graph $H^k$ is the...
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| Other Authors: | , , , |
| Format: | article |
| Language: | eng |
| Published: |
2017
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| Subjects: | |
| Online Access: | http://hdl.handle.net/10773/18640 |
| Country: | Portugal |
| Oai: | oai:ria.ua.pt:10773/18640 |
| Summary: | For a (simple) graph $H$ and non-negative integers $c_0,c_1,\ldots,c_d$ ($c_d \neq 0$), $p(H)=\sum_{k=0}^d{c_k \cdot H^k}$ is the lexicographic polynomial in $H$ of degree $d$, where the sum of two graphs is their join and $c_k \cdot H^k$ is the join of $c_k$ copies of $H^k$. The graph $H^k$ is the $k$th power of $H$ with respect to the lexicographic product ($H^0 = K_1$). The spectrum (if $H$ is regular) and the Laplacian spectrum (in general case) of $p(H)$ are determined in terms of the spectrum of $H$ and~$c_k$'s. Constructions of infinite families of cospectral or integral graphs are also announced. |
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