图$(\bigcup\limits_{i{\in}A}P_i){\bigcup}(\bigcup\limits_{j{\in}B}U_j)$伴随唯一的充要条件

Necessary and Sufficient Condition for Adjoint Uniqueness of the Graph $(\bigcup_{i{\in}A}P_i){\bigcup}(\bigcup_{j{\in}B}U_j)$

中文关键词:  伴随唯一  最小实根  色唯一.

英文关键词:adjointly unique  minimum real root  chromatically unique.

中文摘要:

      设$h(G; x) =h(G)$和$[G]_h$分别表示图$G$的伴随多项式和伴随等价类. 文中给出了$[G]_h$的一个新应用. 利用$[G]_h$, 给出了图$H{\;}(H \cong G)$伴随唯一的充要条件, 其中$H=(\bigcup_{i{\in}A}P_i){\bigcup}(\bigcup_{j{\in}B}U_j)$, $A \subseteq A^{'}=\{1,2,3,5\} \bigcup \{2n|n \in N, n \geq 3\}$, $B \subseteq B^{'}

英文摘要:

      For a graph $G$, let $h(G;x)=h(G)$ and $[G]_h$ denote the adjoint polynomial and the adjoint equivalence class of G, respectively. In this paper, a new application of $[G]_h$ is given. Making use of $[G]_h$, we give a necessary and suffcient condition for adjoint uniqueness of the graph $H$ such that $H\neq G$, where $H=(\bigcup_{i{\in}A}P_i){\bigcup}(\bigcup_ {j{\in}B}U_j)$, $A \subseteq A'=\{1,2,3,5\} \bigcup \{2n|n \in N, n \geq 3\}$, $B \subseteq B'= \{7,2n|n \in N, n \geq 5\}$ and $G=aP_1{\bigcup}a_0P_2{\bigcup}a_1P_3{\bigcup}a_2P_5{\bigcup}({\bigcup}_{i=3}^na_iP_{2i})$.