若干图类的测地Leech标号研究
On Geodesic Leech Labeling of Some Graph Classes
AI总结：
该文研究图的测地Leech标号问题，证明n≥5的圈为非测地Leech图、正则完全二部图中至多3个是测地Leech图，指出度序列无法刻画该类图，还求出轮图的测地路径数并给出W₅、W₆的对应标号。
摘要：
设$f:E\rightarrow \{1,2,3,\dots\}$为图$G$的边标号。图$G$的测地路径数$t_{gp}(G)$是$G$中测地路径的数量。若图$G$中所有测地路径的权重构成集合$\{1,2,3,\dots,t_{gp}(G)\}$（其中路径$P$的权重为其所有边的标号之和），则边标号$f$被称为测地Leech标号。存在测地Leech标号的图称为测地Leech图，否则称为非测地Leech图。本文证明了$n \geq 5$的圈$C_n$均为非测地Leech图；还证明了正则完全二部图中至多有3个是测地Leech图；研究表明度序列无法刻画测地Leech图。此外，本文求出了轮图$W_n$的测地路径数，并给出了$W_5$和$W_6$的测地Leech标号构造。
英文摘要：
Let $f:E\rightarrow \{1,2,3,\dots\}$ be an edge labeling of $G$. The geodesic path number of $G$, $t_{gp}(G)$, is the number of geodesic paths in $G$. An edge labeling $f$ is called a geodesic Leech labeling, if the set of weights of the geodesic paths in $G$ is $\{1,2,3,\dots,t_{gp}(G)\}$, where the weight of a path $P$ is the sum of the labels assigned to the edges of $P$. A graph which admits a geodesic Leech labeling is called a geodesic Leech graph. Otherwise, we call it a non-geodesic Leech graph. In this paper, we prove that cycles $C_n$, $n \geq 5$ are non-geodesic Leech graphs. We also prove that there are at most three regular complete bipartite graphs that are geodesic Leech. We show that degree sequence cannot characterize geodesic Leech graphs. The geodesic path number of the wheel graph $W_n$ is obtained and the geodesic Leech labeling of $W_5$ and $W_6$ is given.
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