论文标题
建造新的传输不规则图
Constructing new families of transmission irregular graphs
论文作者
论文摘要
图$ g $的顶点$ v $的传输是从$ v $到$ g $的所有其他顶点的距离。如果其所有顶点都具有成对的不同传输，则图形是不规则的。一棵类似星的树$ t（k_1，\ ldots，k_t）$是通过连接到隔离的顶点$ t $ t $ pendant路径$ k_1，\ ldots，k_t $获得的树。可以证明，如果一棵类似星的树$ t（a，a+1，\ ldots，a+k）$，$ k \ ge 2 $是奇数的，那么它是不规则的。 $ t（1,2，\ ldots，\ ell）$，$ \ ell \ ge 3 $在且仅当$ \ ell \ ell \ notin \ {r^2 + 1：\ r \ g ge 2 \} $时，才会不规则地传输。确定了星形树木和双星般的树木中的其他无限家庭。传输不规则的独立无限家庭，尤其是$ t（a，a+1，a+2）$，$ a \ ge 2 $的线图在且仅当$ a $均为$时都不规则。
The transmission of a vertex $v$ of a graph $G$ is the sum of distances from $v$ to all the other vertices in $G$. A graph is transmission irregular if all of its vertices have pairwise different transmissions. A starlike tree $T(k_1,\ldots,k_t)$ is a tree obtained by attaching to an isolated vertex $t$ pendant paths of lengths $k_1,\ldots,k_t$, respectively. It is proved that if a starlike tree $T(a,a+1,\ldots,a+k)$, $k\ge 2$, is of odd order, then it is transmission irregular. $T(1,2,\ldots,\ell)$, $\ell \ge 3$, is transmission irregular if and only if $\ell \notin \{r^2 + 1:\ r\ge 2\}$. Additional infinite families among the starlike trees and bi-starlike trees are determined. Transmission irregular unicyclic infinite families are also presented, in particular, the line graph of $T(a,a+1,a+2)$, $a\ge 2$, is transmission irregular if and only if $a$ is even.
