论文标题
在带有路径的植根产品的行走矩阵决定因素上的猜想的证明
Proof of a conjecture on the determinant of walk matrix of rooted product with a path
论文作者
论文摘要
$ n $ vertex Graph $ g $带有邻接矩阵$ a $的步行矩阵，由$ w（g）$表示，是$ [e，ae，ae，\ ldots，a^{n-1} e] $，其中$ e $是全部矢量。令$ g \ circ p_m $为$ g $的根产品和扎根的路径$ p_m $（以eNdvertex为根），即，$ g \ circ p_m $是从$ g $和$ n $ copies of $ p_m $ $ g $ a $ g $的$ g $ a Compoy of $ p_m $ p_m $ p_m $ p_m $ p_m $ p_m获得的图表。 Mao-Liu-Wang（2015）和Mao-Wang（2022）证明，$ m = 2 $和$ m \ in \ {3,4 \} $，$ \ det w（g \ circ p_m）= \ circ p_m）= \ pm a_0^{ w（g））^m，$$，其中$ a_0 $是$ g $的特征多项式的恒定术语。此外，Mao-Wang（2022）猜想该公式适用于任何$ m \ ge 2 $。在本说明中，我们使用Chebyshev多项式技术来验证这种猜想。
The walk matrix of an $n$-vertex graph $G$ with adjacency matrix $A$, denoted by $W(G)$, is $[e,Ae,\ldots,A^{n-1}e]$, where $e$ is the all-ones vector. Let $G\circ P_m$ be the rooted product of $G$ and a rooted path $P_m$ (taking an endvertex as the root), i.e., $G\circ P_m$ is a graph obtained from $G$ and $n$ copies of $P_m$ by identifying each vertex of $G$ with an endvertex of a copy of $P_m$. Mao-Liu-Wang (2015) and Mao-Wang (2022) proved that, for $m=2$ and $m\in\{3,4\}$, respectively$$\det W(G\circ P_m)=\pm a_0^{\lfloor\frac{m}{2}\rfloor}(\det W(G))^m,$$ where $a_0$ is the constant term of the characteristic polynomial of $G$. Furthermore, Mao-Wang (2022) conjectured that the formula holds for any $m\ge 2$. In this note, we verify this conjecture using the technique of Chebyshev polynomials.
