Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: In the main part of this paper  present polynomial expressions for the cardinalities of some sets of interest of the nice distance-layer structure of the ll-known De Bruijn and Kautz digraphs. More precisely, given a vertex $v$, let  $S_{i}^\star(v)$ be the set of vertices  at distance $i$ from $v$.  show that $|S_{i}^\star(v)|=d^i-a_{i-1}d^{i-1}-\cdots -a_{1} d-a_{0}$, where $d$ is the degree of the digraph and  the coefficients $a_{k}\in\{0,1\}$ are explicitly  calculated. Analogously, let $w$ be a vertex adjacent from $v$ such that $S_{i}^\star(v)\cap S_j^{\ast}(w)\neq \emptyset$ for some $j$.  prove that $\big |S_{i}^\star(v) \cap S_j^{\ast}(w) \big |=d^i-b_{i-1}d^{i-1}-\ldots -b_{1} d-b_{0},$ where the coefficients $b_{t}\in\{0,1\}$ are determined from the coefficients $a_k$ of the polynomial expression of $|S_{i}^\star(v)|$. An application to deflection routing in De Bruijn and Kautz networks serves as motivation for our study. It is worth-mentioning that our analysis can be extended to other families of digraphs on alphabet or to general iterated line digraphs.
['Josep M. Fabrega', 'Jaume Martí-Farré', 'Xavier Muñoz']
I am writing a research paper on a idea Layer structure of De Bruijn and Kautz digraphs: an application to deflection routing
[{"from": "human", "value": "I am writing a research paper on a idea Layer structure of De Bruijn and Kautz digraphs: an application to deflection routing"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: In the main part of this paper  present polynomial expressions for the cardinalities of some sets of interest of the nice distance-layer structure of the ll-known De Bruijn and Kautz digraphs. More precisely, given a vertex $v$, let  $S_{i}^\\star(v)$ be the set of vertices  at distance $i$ from $v$.  show that $|S_{i}^\\star(v)|=d^i-a_{i-1}d^{i-1}-\\cdots -a_{1} d-a_{0}$, where $d$ is the degree of the digraph and  the coefficients $a_{k}\\in\\{0,1\\}$ are explicitly  calculated. Analogously, let $w$ be a vertex adjacent from $v$ such that $S_{i}^\\star(v)\\cap S_j^{\\ast}(w)\\neq \\emptyset$ for some $j$.  prove that $\\big |S_{i}^\\star(v) \\cap S_j^{\\ast}(w) \\big |=d^i-b_{i-1}d^{i-1}-\\ldots -b_{1} d-b_{0},$ where the coefficients $b_{t}\\in\\{0,1\\}$ are determined from the coefficients $a_k$ of the polynomial expression of $|S_{i}^\\star(v)|$. An application to deflection routing in De Bruijn and Kautz networks serves as motivation for our study. It is worth-mentioning that our analysis can be extended to other families of digraphs on alphabet or to general iterated line digraphs."}]