arXiv subjects

Sung-Yell Song

Publications and source records attributed to Sung-Yell Song.

4 recordsLinked to original sources

arXiv2022

Partial geometric designs having circulant concurrence matrices

We survey partial geometric designs and investigate their concurrences of points. The concurrence matrix of a design, which encodes the concurrences of pairs of points, can be used in the classification of designs in some extent. An ordinary 2-$(v,k,λ)$ design has concurrence $λ$ for any pair of distinct points, and its concurrence matrix is circulant. A partial geometry has two concurrences $1$ and $0$ and a transversal design TD$_λ(k, u)$ has two concurrences $λ$ and $0$. It is also known that the concurrence matrix of a partial geometric design can have at most three distinct eigenvalues, all of which are non-negative integers. In this paper, we show the existence of other partial geometric designs having two or three distinct concurrences, and investigate which symmetric circulant matrices are realized as the concurrence matrices of partial geometric designs. We collect known sources of partial geometric designs and study their structural characteristics and construction methods. We then give a list of feasible parameter sets for partial geometric designs of order up to 12 each of which has a circulant concurrence matrix. We also consider the combinatorial properties and constructions of partial geometric designs satisfying these parameter sets.

math.CO↗

arXiv2020

Hadamard diagonalizable graphs of order at most 36

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 an efficient computation 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.

math.CO↗

arXiv2018

Graphs that are cospectral for the distance Laplacian

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 $\mathcal{D}(G)$. We establish several general methods for producing $\mathcal{D}^L$-cospectral graphs that can be used to construct infinite families. We provide examples showing that various properties are not preserved by $\mathcal{D}^L$-cospectrality, including examples of $\mathcal{D}^L$-cospectral strongly regular and circulant graphs. We establish that the absolute values of coefficients of the distance Laplacian characteristic polynomial are decreasing, i.e., $|δ^L_{1}|\geq \dots \geq |δ^L_{n}|$ where $δ^L_{k}$ is the coefficient of $x^k$.

math.CO↗

arXiv2017

On the Wiener index, distance cospectrality and transmission regular graphs

In this paper, we investigate various algebraic and graph theoretic properties of the distance matrix of a graph. Two graphs are $D$-cospectral if their distance matrices have the same spectrum. We construct infinite pairs of $D$-cospectral graphs with different diameter and different Wiener index. A graph is $k$-transmission-regular if its distance matrix has constant row sum equal to $k$. We establish tight upper and lower bounds for the row sum of a $k$-transmission-regular graph in terms of the number of vertices of the graph. Finally, we determine the Wiener index and its complexity for linear $k$-trees, and obtain a closed form for the Wiener index of block-clique graphs in terms of the Laplacian eigenvalues of the graph. The latter leads to a generalization of a result for trees which was proved independently by Mohar and Merris.

math.CO↗