Mathematics

CLC number: O 157.5

Some Properties of a Class of Addition Cayley Signed Graph

Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China

^† To whom correspondence should be addressed. E-mail: wangwzh @mail.lzjtu. cn

Received: 10 March 2022

Abstract

Let $G=\mathrm{C}\mathrm{a}{\mathrm{y}}^{+}(Z_{p_{\mathrm{1}}}\times Z_{p_{\mathrm{1}}^{{\alpha }_{\mathrm{1}}}{p}_{\mathrm{2}}^{{\alpha }_{\mathrm{2}}}\cdots p_k^{{\alpha }_k}},\Phi )$ be a simple graph having vertex set $V(G)=Z_{p_{\mathrm{1}}}\times Z_{p_{\mathrm{1}}^{{\alpha }_{\mathrm{1}}}{p}_{\mathrm{2}}^{{\alpha }_{\mathrm{2}}}\cdots p_k^{{\alpha }_k}}$and edge set $E(G)=\{\{x,y\}:x+y\in \Phi \}$, where all $p_{\mathrm{1}},p_{\mathrm{2}},\cdots ,p_k$ are distinct prime factors and $\Phi$ is the set of all units of the ring $Z_{p_{\mathrm{1}}}\times Z_{p_{\mathrm{1}}^{{\alpha }_{\mathrm{1}}}{p}_{\mathrm{2}}^{{\alpha }_{\mathrm{2}}}\cdots p_k^{{\alpha }_k}}.\mathrm{L}\mathrm{e}\mathrm{t}\Sigma =(G,\sigma )$be a signed graph whose underlying graph is $G$ and signature function is $\sigma :E(G)\to \{+\mathrm{1},-\mathrm{1}\}$ defined as $\sigma (\{x,y\})=\{\begin{array}{cc}+\mathrm{1},& \mathrm{i}\mathrm{f}x\in \phi (p_{\mathrm{1}})\mathrm{o}\mathrm{r}y\in \phi (p_{\mathrm{1}}^{{\alpha }_{\mathrm{1}}}{p}_{\mathrm{2}}^{{\alpha }_{\mathrm{2}}}\cdots p_k^{{\alpha }_k});\\ -\mathrm{1},& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}.\end{array}$In this paper, we characterize the balance of $\Sigma$ and some graphs derived from it such as $\eta (\Sigma )$, $L(\Sigma )$ and $C_E(\Sigma )$. Moreover, we investigate the clusterability and sign-compativility of $\Sigma$.