arXiv subjects
Publications and source records attributed to Sunyo Moon.
We characterize when the spectral variation of the signed Laplacian matrices is integral after a new edge is added to a signed graph. As an application, for every fixed signed complete graph, we fully characterize the class of signed graphs to which one can recursively add new edges keeping spectral integral variation to make the signed complete graph.
Let $G$ be a unicyclic graph. In this paper, we provide an upper bound for the number of Laplacian eigenvalues of $G$ within the interval $[0,1)$ in terms of the diameter and the girth of $G$.
We present lower and upper bounds for the geometric-arithmetic index of unicyclic graphs and provide extremal graphs for the corresponding bounds.
For some positive integer $k$, if the finite cyclic group $\mathbb{Z}_k$ can act freely on a graph $G$, then we say that $G$ is $k$-symmetric. In 1985, Faria showed that the multiplicity of Laplacian eigenvalue 1 is greater than or equal to the difference between the number of pendant vertices and the number of quasi-pendant vertices. But if a graph has a pendant vertex, then it is at most 1-connected. In this paper, we investigate a class of 2-connected $k$-symmetric graphs with a Laplacian eigenvalue 1. We also identify a class of $k$-symmetric graphs in which all Laplacian eigenvalues are integers.
In this paper, we provide an explicit formula for the rank of the walk matrix of the extended Dynkin graph $\tilde{D}_n$.
Let $m_GI$ denote the number of Laplacian eigenvalues of a graph $G$ in an interval $I$ and let $\alpha(G)$ denote the independence number of $G$. In this paper, we determine the classes of graphs that satisfy the condition $m_G[0,n-\alpha(G)]=\alpha(G)$ when $\alpha(G)= 2$ and $\alpha(G)= n-2$, where $n$ is the order of $G$. When $\alpha(G)=2$, $G \cong K_1 \nabla K_{n-m} \nabla K_{m-1}$ for some $m \geq 2$. When $\alpha(G)=n-2$, there are two types of graphs $B(p,q,r)$ and $B'(p,q,r)$ of order $n=p+q+r+2$, which we call the binary star graphs. Also, we show that the binary star graphs with $p=r$ are determined by their Laplacian spectra.