论文标题
签名图的特征值
Eigenvalues of signed graphs
论文作者
论文摘要
签名图的边缘标记为正或负数。 $ρ（m）$表示$ m $ -spectral半径为$σ$，其中$ m = m（σ）$是$σ$的真实对称图矩阵。显然，$ρ（m）= \ mbox {max} \ {λ_1（m），-λ_n（m）\} $。令$ a（σ）$为$σ$和$（k_n，h^ - ）$的邻接矩阵是一个签名的完整图，其负边缘会导致a子图$ h $。在本文中，我们首先关注光谱极端图理论中的中心问题，如下所示：在$（k_n，t^ - ）$中，哪个签名图具有最大$ρ（a（σ））$，其中$ t $是$ t $？为了回答问题，我们分别在$（k_n，t^ - ）$中分别表征了最大$λ_1（a（σ））$和最小$λ_n（a（σ））$的极端签名图。签名图的另一个有趣的图矩阵是距离矩阵，即由Hameed，Shijin，Shijin，Soorya，Germina和Zaslavsky定义的$ D（σ）$ [8]。请注意，$ a（σ）= d（σ）$当$σ\ in（k_n，t^ - ）$。在本文中，我们在符号图$σ$的最小距离特征值上给出了至少2个。这意味着Lin [11]证明的结果最初是由Aouchiche和Hansen [1]猜想的。
Signed graphs have their edges labeled either as positive or negative. $ρ(M)$ denote the $M$-spectral radius of $Σ$, where $M=M(Σ)$ is a real symmetric graph matrix of $Σ$. Obviously, $ρ(M)=\mbox{max}\{λ_1(M),-λ_n(M)\}$. Let $A(Σ)$ be the adjacency matrix of $Σ$ and $(K_n,H^-)$ be a signed complete graph whose negative edges induce a subgraph $H$. In this paper, we first focus on a central problem in spectral extremal graph theory as follows: Which signed graph with maximum $ρ(A(Σ))$ among $(K_n,T^-)$ where $T$ is a spanning tree? To answer the problem, we characterize the extremal signed graph with maximum $λ_1(A(Σ))$ and minimum $λ_n(A(Σ))$ among $(K_n,T^-)$, respectively. Another interesting graph matrix of a signed graph is distance matrix, i.e. $D(Σ)$ which was defined by Hameed, Shijin, Soorya, Germina and Zaslavsky [8]. Note that $A(Σ)=D(Σ)$ when $Σ\in (K_n,T^-)$. In this paper, we give upper bounds on the least distance eigenvalue of a signed graph $Σ$ with diameter at least 2. This result implies a result proved by Lin [11] was originally conjectured by Aouchiche and Hansen [1].
