There are four prominent product graphs in graph theory: Cartesian, strong, direct, and lexicographic. Of these four product graphs, the lexicographic product graph is the least studied. Lexicographic products are not commutative but still have some interesting properties. This paper begins with basic definitions of graph theory, including the definition of a graph, that are needed to understand theorems and proofs that come later. The paper then discusses the lexicographic product of digraphs, denoted $G circ H$, for some digraphs $G$ and $H$. The paper concludes by proving a cancellation property for the lexicographic product of digraphs $G$, $H$, $A$, and $B$: if $G circ H cong A circ B$ and $|V(G)| = |V(A)|$, then $G cong A$. It also proves additional cancellation properties for lexicographic product digraphs and the author hopes the final result will provide further insight into tournaments.
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机译：图论中有四个突出的乘积图：笛卡尔，强，直接和词典。在这四个产品图中，字典词典产品图最少。词典产品不是可交换的，但仍具有一些有趣的属性。本文从图论的基本定义开始，包括图的定义，这是理解后面的定理和证明所必需的。然后，本文讨论了有向图$ G $和$ H $的有向图的词典产品，表示为$ G circ H $。本文最后证明了有向图$ G $，$ H $，$ A $和$ B $的词典产品的抵消性质：如果$ G circ H cong A circ B $和$ | V（G ）| = | V（A）| $，然后是$ G cong A $。它还证明了词典产品有向图的其他取消属性，并且作者希望最终结果将提供对比赛的进一步了解。
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