arXiv 2212.116432022-12-22math.CO

图矩阵特征值重数的电路秩定界新方法

Bounding the multiplicities of eigenvalues of graph matrices in terms of circuit rank using a new approach

Ahmet Batal

摘要：

设$G$是一个简单无向图，$θ(G)$是$G$的圈秩，$η_M(G)$和$m_M(G,λ)$分别是图矩阵$M(G)$的特征值$λ$的零值和重值.在$M(G)$是邻接矩阵$A(G)$的情况下，（拉普拉斯矩阵$L(G)$，无符号拉普拉斯矩阵$Q(G)$），当$λ$是整数（分别为偶数）时，我们根据$θ(G)$找到$m_M(G,λ)$的界限。我们还表明，当$α$和$λ$是有理数类似的界限可以找到$m_{A_α}(G,λ)$，其中$A_α(G)$是广义邻接矩阵的$G$。我们的边界只包含$θ(G)$，而不是它的倍数。到目前为止，只有$m_A(G,λ)$（以及后来的$m_{A_α}(G,λ)$）的边界已经被发现在电路秩方面，它们都包含$2θ(G)$。只有一个例外，那就是$λ=0$。Wong等人（2022）证明了$η_A(G_c)\leq θ(G_c)+1$，其中$G_c$是一个连通的仙人掌，其块是偶数圈。我们的结果，特别是，推广和扩展这一结果的任何偶连通图$G$的A（G）的任何偶特征值的重数，以及任何连通图$G$的任何偶特征值的$L(G)$和$Q(G)$。他们还表明，$η_A(G_c)\leq 1$时，每一个块的仙人掌是一个奇数周期。这也对应于我们界的一个特例。

英文摘要：

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.

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英文摘要：

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