关于距离谱间隙与D-等能图的构造
On the distance spectral gap and construction of D-equienergetic graphs
AI总结：
本文研究图的距离谱间隙，得到其基于距离特征值平方和的上下界，给出距离特征值与距离能量的界，并构造了直径为3和4的非D-同谱D-等能图新家族。
摘要：
设$D(G)$表示具有$n$个顶点的连通图$G$的距离矩阵。图$G$的距离谱间隙定义为$δ_{D^G} = ρ_1 - ρ_2$，其中$ρ_1$和$ρ_2$分别表示$D(G)$的最大和第二大特征值。对于$k$-传递正则图$G$，距离拉普拉斯矩阵的第二小特征值等于$G$的距离谱间隙。本文我们根据图的距离特征值平方和，得到了图距离谱间隙的一些上界和下界。此外，我们还给出了图的距离特征值和距离能量的一些界。进一步，我们构造了直径为3和4的非D-同谱D-等能图的新家族。
英文摘要：
Let $D(G)$ denote the distance matrix of a connected graph $G$ with $n$ vertices. The distance spectral gap of a graph $G$ is defined as $δ_{D^G} = ρ_1 - ρ_2$, where $ρ_1$ and $ρ_2$ represent the largest and second largest eigenvalues of $D(G)$, respectively. For a $k$-transmission regular graph $G$, the second smallest eigenvalue of the distance Laplacian matrix equals the distance spectral gap of $G$. In this article, we obtain some upper and lower bounds for the distance spectral gap of a graph in terms of the sum of squares of its distance eigenvalues. Additionally, we provide some bounds for the distance eigenvalues and distance energy of graphs. Furthermore, we construct new families of non $D$-cospectral $D$-equienergetic graphs with diameters of $3$ and $4$.
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