AI 中文总结
研究关于路径\(P_n\)的不规则图，证明对于\(k \geq 4\)存在无限族图对\(4 \leq n \leq k\)及\(P_n\)各根\(r\)同时满足\(P_n\)-不规则和\((P_n)_r\)-不规则，还针对\(P_3\)给出不同根情况下的结果，证实相关强猜想。
AI 中文摘要
设\(P_n\)表示\(n\)个顶点的路径。若简单有限图\(G\)中任意两个不同顶点属于不同数量的同构于\(P_n\)的子图，则称\(G\)为\(P_n\)-不规则的。对于\(P_n\)的固定顶点\(r\)（根），若\(G\)中任意两个不同顶点在同构于\(P_n\)的子图中作为根\(r\)的数量不同，则称\(G\)为\((P_n)_r\)-不规则的。本文证明，对于每个整数\(k \geq 4\)，存在无限族图，对于满足\(4 \leq n \leq k\)的每个整数\(n\)以及\(P_n\)的每个根\(r\)，这些图同时是\(P_n\)-不规则和\((P_n)_r\)-不规则的。对于路径\(P_3\)，若\(r\)是中心顶点，则不存在非平凡的\((P_3)_r\)-不规则图；若\(r\)是\(P_3\)的端点，则构造了无限多个既是\(P_3\)-不规则又是\((P_3)_r\)-不规则的图。特别地，这些结果证实了关于\(F\)-不规则图的强猜想在\(F\)为路径\(P_n\)的情况下成立。
英文摘要
Let $P_n$ denote a path on $n$ vertices. A simple finite graph $G$ is called $P_n$-irregular if any two distinct vertices of $G$ belong to a different number of subgraphs of $G$ isomorphic to $P_n$. Alternatively, for a fixed vertex $r$ of $P_n$ (the root), $G$ is called $(P_n)_r$-irregular if any two distinct vertices of $G$ act as the root $r$ in a different number of subgraphs of $G$ isomorphic to $P_n$. This paper proves that for each integer $k \geq 4$, there exists an infinite family of graphs that are simultaneously $P_n$-irregular and $(P_n)_r$-irregular for every integer $n$ satisfying $4 \leq n \leq k$ and every root $r$ of $P_n$. For the path $P_3$, we observe that no nontrivial $(P_3)_r$-irregular graphs exist if $r$ is the central vertex. In contrast, if $r$ is an end-vertex of $P_3$, an infinite collection of graphs is constructed that are both $P_3$-irregular and $(P_3)_r$-irregular. In particular, these results confirm the Strong Conjecture about $F$-irregular graphs for the case where $F$ is a path $P_n$.
Comments29 pages, 4 figures
