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Creators/Authors contains: "Song, Sung-Yell"

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  1. ; ; ; ; ; ; ; ; ; (, The Electronic Journal of Combinatorics)
    If the Laplacian matrix of a graph has a full set of orthogonal eigenvectors with entries $$\pm1$$, then the matrix formed by taking the columns as the eigenvectors is a Hadamard matrix and the graph is said to be Hadamard diagonalizable. In this article, we prove that if $n=8k+4$ the only possible Hadamard diagonalizable graphs are $$K_n$$, $$K_{n/2,n/2}$$, $$2K_{n/2}$$, and $$nK_1$$, and we develop a computational method for determining all graphs diagonalized by a given Hadamard matrix of any order. Using these two tools, we determine and present all Hadamard diagonalizable graphs up to order 36. Note that it is not even known how many Hadamard matrices there are of order 36. 
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  2. ; ; ; ; ; ; (, The Electronic Journal of Linear Algebra)
    null (Ed.)
    The distance matrix $$\mathcal{D}(G)$$ of a graph $$G$$ is the matrix containing the pairwise distances between vertices, and the distance Laplacian matrix is $$\mathcal{D}^L (G)=T(G)-\mathcal{D} (G)$$, where $T(G)$ is the diagonal matrix of row sums of $$\mathcal{D}(G)$$. Several general methods are established for producing $$\mathcal{D}^L$$-cospectral graphs that can be used to construct infinite families. Examples are provided to show that various properties are not preserved by $$\mathcal{D}^L$$-cospectrality, including examples of $$\mathcal{D}^L$$-cospectral strongly regular and circulant graphs. It is established that the absolute values of coefficients of the distance Laplacian characteristic polynomial are decreasing, i.e., $$|\delta^L_{1}|\geq \cdots \geq |\delta^L_{n}|$$, where $$\delta^L_{k}$$ is the coefficient of $x^k$. 
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