论文标题
环$ C_ \ Mathscr {p}（x）$的歼灭仪图
Annihilator graph of the ring $C_\mathscr{P}(X)$
论文作者
论文摘要
在本文中，我们介绍了环$ C_ \ Mathscr {p}（x）$的歼灭仪图，由$ ag（C_ \ Mathscr {p}（x））$表示，并观察基础Tychonoff Space $ X $在$ AG（C_ \ Mathscr of ag Mathscr的各种图形上）的效果。通常，$ ag（c_ \ Mathscr {p}（x））$，通常位于零的除数图和弱零的零除法图之间，$ c_ \ mathscr {p}（x）$，证明这三个图是在所有$ \ nath $ scr $ points $ points $ points $ points $ points $ x $} $ x $} $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x {p时仅重合的 - $ \ leq 2 $。确定合适的$ ag（C_ \ Mathscr {p}（x））$的合适的子图，称为$ g（c_ \ Mathscr {p}（x））$，我们确定$ ag（C_ \ Mathscr {p}（p}（x）（x）（x）（x）（x）（x））$ g（c_ \ nath proparies and promore and promore and atemertial and apparial and ate and ate and ate ate ate ate ate ate ate ate ator a promore a promore ate ator a promore，例如，直径，偏心，周长，半径，色数和集团数。通过选择环$ c_ \ mathscr {p}（x）$，其中$ \ mathscr {p} $是$ x $的所有有限子集的理想之处，使得$ x_ \ mathscr {p} $是有限的，我们为$ g（c_ \ mathscr n of then then then then then $ g} $ ged $ get n of normumantic and of mathscr normumantion and $ ged ged and per n of get per n of ch o y n of。 $ ag（c_ \ mathscr {p}（x））$。这仅通过有限数量的颜色来表现出无限图形的实例。我们表明，任何图的同构$ψ：ag（c_ \ Mathscr {p}（x））\ rightarrow ag（c_ \ Mathscr {q}（y））$ maps $ g（c_ \ Mathscr {p}（p}（x）） $ ϕ：g（c_ \ mathscr {p}（x））\ rightArrow g（c_ \ mathscr {q}（y））$可以扩展到图形等法$ψ：ag（c_ \ mathscr {p}（p}（p}（x）（x）（x））\ rightArrow ag（c_ rightArrow ag（c_ \ rightarrow ag（c_ \ mathsscriftiction）最后，我们表明，对于$ c_ \ mathscr {p}（x）$，至少有许多$ \ mathscr {p} $ - 点，就图形属性而言，诱导的子graph $ g（c_ \ mathscr {p}（x））$是$ ag（c_ mathsscr $ ag（c_ math）的良好替代品。
In this article, we introduce the annihilator graph of the ring $C_\mathscr{P}(X)$, denoted by $AG(C_\mathscr{P}(X))$ and observe the effect of the underlying Tychonoff space $X$ on various graph properties of $AG(C_\mathscr{P}(X))$. $AG(C_\mathscr{P}(X))$, in general, lies between the zero divisor graph and weakly zero divisor graph of $C_\mathscr{P}(X)$ and it is proved that these three graphs coincide if and only if the cardinality of the set of all $\mathscr{P}$-points, $X_\mathscr{P}$ is $\leq 2$. Identifying a suitable induced subgraph of $AG(C_\mathscr{P}(X))$, called $G(C_\mathscr{P}(X))$, we establish that both $AG(C_\mathscr{P}(X))$ and $G(C_\mathscr{P}(X))$ share similar graph theoretic properties and have the same values for the parameters, e.g., diameter, eccentricity, girth, radius, chromatic number and clique number. By choosing the ring $C_\mathscr{P}(X)$ where $\mathscr{P}$ is the ideal of all finite subsets of $X$ such that $X_\mathscr{P}$ is finite, we formulate an algorithm for coloring the vertices of $G(C_\mathscr{P}(X))$ and thereby get the chromatic number of $AG(C_\mathscr{P}(X))$. This exhibits an instance of coloring infinite graphs by just a finite number of colors. We show that any graph isomorphism $ψ: AG(C_\mathscr{P}(X)) \rightarrow AG(C_\mathscr{Q}(Y))$ maps $G(C_\mathscr{P}(X))$ isomorphically onto $G(C_\mathscr{Q}(Y))$ as a graph and a graph isomorphism $ϕ: G(C_\mathscr{P}(X)) \rightarrow G(C_\mathscr{Q}(Y))$ can be extended to a graph isomorphism $ψ: AG(C_\mathscr{P}(X)) \rightarrow AG(C_\mathscr{Q}(Y))$ under a mild restriction on the function $ϕ$. Finally, we show that atleast for the rings $C_\mathscr{P}(X)$ with finitely many $\mathscr{P}$-points, so far as the graph properties are concerned, the induced subgraph $G(C_\mathscr{P}(X))$ is a good substitute for $AG(C_\mathscr{P}(X)$.
