论文标题
使用一种新方法来界定图形矩阵的特征值的多重性
Bounding the multiplicities of eigenvalues of graph matrices in terms of circuit rank using a new approach
论文作者
论文摘要
令$ g $为简单的无向图，$θ（g）$是$ g $，$η_m（g）$和$ m_m（g，λ）$的电路排名，分别为无效和多样化图形矩阵$ m（g）$的eigenvalue $λ$。在情况下，$ m（g）$是邻接矩阵$ a（g）$（laplacian矩阵$ l（g）$，无标志性的laplacian矩阵$ q（g）$），我们在$θ（g）$λ$时（g）$λ$ nisteger（甚至是integer），在$θ（g）上，我们找到范围为$ m_m（g，λ）$。我们还表明，当$α$和$λ$是有理数时，可以找到$ m_ {a_α}（g，λ）$的相似边界，其中$a_α（g）$是$ g $的广义邻接矩阵。我们的边界仅包含$θ（g）$，而不是其中的倍数。到目前为止，只有$ m_a（g，λ）$（和后来的$ m_ {a_α}（g，λ）$）的界限已被发现，而这些电路等级的限制为$2θ（g）$。在$λ= 0 $的情况下，只有一个例外。 Wong等。 （2022）表明$η_a（g_c）\leqθ（g_c）+1 $，其中$ g_c $是一个连接的仙人掌，其块甚至是周期。我们的结果，尤其是将此结果推广到任何连接的图形$ g $的a（g）的偶数均匀性，以及任何连接的$ l（g）$和$ q（g）$的任何均匀的特征值。他们还表明，当仙人掌的每个街区都是一个奇怪的周期时，$η_a（g_c）\ leq 1 $。这也对应于我们界限的特殊情况。
Let $G$ be a simple undirected graph, $θ(G)$ be the circuit rank of $G$, $η_M(G)$ and $m_M(G,λ)$ be the nullity and the multiplicity of eigenvalue $λ$ of a graph matrix $M(G)$, respectively. In the case $M(G)$ is the adjacency matrix $A(G)$, (the Laplacian matrix $L(G)$, the signless Laplacian matrix $Q(G)$) we find bounds to $m_M(G,λ)$ in terms of $θ(G)$ when $λ$ is an integer (even integer, respectively). We also show that when $α$ and $λ$ are rational numbers similar bounds can be found for $m_{A_α}(G,λ)$ where $A_α(G)$ is the generalized adjaceny matrix of $G$. Our bounds contain only $θ(G)$, not a multiple of it. Up to now only bounds of $m_A(G,λ)$ (and later $m_{A_α}(G,λ)$) have been found in terms of the circuit rank and all of them contains $2θ(G)$. There is only one exception in the case $λ=0$. Wong et al. (2022) showed that $η_A(G_c)\leq θ(G_c)+1$, where $G_c$ is a connected cactus whose blocks are even cycles. Our result, in particular, generalizes and extends this result to the multiplicity of any even eigenvalue of A(G) of any even connected graph $G$, and of any even eigenvalue of $L(G)$ and $Q(G)$ of any connected graph $G$. They also showed that $η_A(G_c)\leq 1$ when every block of the cactus is an odd cycle. This also corresponds a special case of our bound.
