Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: An embedding $i \mapsto p_i\in \mathbb{R}^d$ of the vertices of a graph $G$ is called universally completable if the following holds: For any other embedding $i\mapsto q_i~\in \mathbb{R}^{k}$ satisfying $q_i^T q_j = p_i^T p_j$ for $i = j$ and $i$ adjacent to $j$, there exists an isometry mapping the $q_i$'s to the $p_i$'s for all $ i\in V(G)$. The notion of universal completability was introduced recently due to its relevance to the positive semidefinite matrix completion problem. In this work  focus on graph embeddings constructed using the eigenvectors of the least eigenvalue of the adjacency matrix of $G$, which  call least eigenvalue frameworks.  identify two necessary and sufficient conditions for such frameworks to be universally completable. Our conditions also allow us to give algorithms for determining whether a least eigenvalue framework is universally completable. Furthermore, our computations for Cayley graphs on $\mathbb{Z}_2^n \ (n \le 5)$ show that almost all of these graphs have universally completable least eigenvalue frameworks. In the second part of this work  study uniquely vector colorable (UVC) graphs, i.e., graphs for which the semidefinite program corresponding to the Lov\'asz theta number (of the complementary graph) admits a unique optimal solution.  identify a sufficient condition for showing that a graph is UVC based on the universal completability of an associated framework. This allows us to prove that Kneser and $q$-Kneser graphs are UVC. Lastly,  show that least eigenvalue frameworks of 1-walk-regular graphs always provide optimal vector colorings and furthermore,  are able to characterize all optimal vector colorings of such graphs. In particular,  give a necessary and sufficient condition for a 1-walk-regular graph to be uniquely vector colorable.
['Chris D. Godsil', 'David E. Roberson', 'Brendan Rooney', 'Robert Šámal', 'Antonios Varvitsiotis']
I am writing a research paper on a idea Universal completability, least eigenvalue frameworks, and vector colorings
[{"from": "human", "value": "I am writing a research paper on a idea Universal completability, least eigenvalue frameworks, and vector colorings"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: An embedding $i \\mapsto p_i\\in \\mathbb{R}^d$ of the vertices of a graph $G$ is called universally completable if the following holds: For any other embedding $i\\mapsto q_i~\\in \\mathbb{R}^{k}$ satisfying $q_i^T q_j = p_i^T p_j$ for $i = j$ and $i$ adjacent to $j$, there exists an isometry mapping the $q_i$'s to the $p_i$'s for all $ i\\in V(G)$. The notion of universal completability was introduced recently due to its relevance to the positive semidefinite matrix completion problem. In this work  focus on graph embeddings constructed using the eigenvectors of the least eigenvalue of the adjacency matrix of $G$, which  call least eigenvalue frameworks.  identify two necessary and sufficient conditions for such frameworks to be universally completable. Our conditions also allow us to give algorithms for determining whether a least eigenvalue framework is universally completable. Furthermore, our computations for Cayley graphs on $\\mathbb{Z}_2^n \\ (n \\le 5)$ show that almost all of these graphs have universally completable least eigenvalue frameworks. In the second part of this work  study uniquely vector colorable (UVC) graphs, i.e., graphs for which the semidefinite program corresponding to the Lov\\'asz theta number (of the complementary graph) admits a unique optimal solution.  identify a sufficient condition for showing that a graph is UVC based on the universal completability of an associated framework. This allows us to prove that Kneser and $q$-Kneser graphs are UVC. Lastly,  show that least eigenvalue frameworks of 1-walk-regular graphs always provide optimal vector colorings and furthermore,  are able to characterize all optimal vector colorings of such graphs. In particular,  give a necessary and sufficient condition for a 1-walk-regular graph to be uniquely vector colorable."}]