@article{article_453103, title={Some properties of the total graph and regular graph of a commutative ring}, journal={Hacettepe Journal of Mathematics and Statistics}, volume={47}, pages={835–843}, year={2018}, author={Ghanem, Manal and Nazzal, Khalida}, keywords={Total graph of a commutative ring,Regular graph of a commutative ring,Locally connected,Locally homogeneous,Regular graph,Eulerian graph}, abstract={Let $R$ be a commutative ring with unity. The total graph of $R$, $T(\Gamma(R))$, is the simple graph with vertex set $R$ and two distinct vertices are adjacent if their sum is a zero-divisor in $R$. Let Reg <span style=’display: inline !important; float: none; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0); font-family: "Helvetica Neue",Helvetica,Arial,sans-serif; font-size: 12.6px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;’>$(\Gamma(R))$ </span> and <span style=’display: inline !important; float: none; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0); font-family: "Helvetica Neue",Helvetica,Arial,sans-serif; font-size: 12.6px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;’>$Z(\Gamma(R))$ </span>be the subgraphs of <span style=’display: inline !important; float: none; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0); font-family: "Helvetica Neue",Helvetica,Arial,sans-serif; font-size: 12.6px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;’>$T(\Gamma(R))$ </span> induced by the set of all regular elements and the set of zero-divisors in $R$, respectively. We determine when each of the graphs <span style=’display: inline !important; float: none; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0); font-family: "Helvetica Neue",Helvetica,Arial,sans-serif; font-size: 12.6px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;’>$T(\Gamma(R))$ </span>, <span style=’display: inline !important; float: none; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0); font-family: "Helvetica Neue",Helvetica,Arial,sans-serif; font-size: 12.6px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;’>Reg </span> <span style="background-color: rgb(255, 255, 255); box-sizing: border-box; color: rgb(0, 0, 0); display: inline; float: none; font-family: "Helvetica Neue",Helvetica,Arial,sans-serif; font-size: 12.6px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;">$(\Gamma(R))$, </span> <span style=’display: inline !important; float: none; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0); font-family: "Helvetica Neue",Helvetica,Arial,sans-serif; font-size: 12.6px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;’> and </span> <span style="background-color: rgb(255, 255, 255); box-sizing: border-box; color: rgb(0, 0, 0); display: inline; float: none; font-family: "Helvetica Neue",Helvetica,Arial,sans-serif; font-size: 12.6px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;">$Z(\Gamma(R))$ </span> is locally connected, and when it is locally homogeneous. When each of <span style=’display: inline !important; float: none; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0); font-family: "Helvetica Neue",Helvetica,Arial,sans-serif; font-size: 12.6px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;’>Reg </span> <span style="background-color: rgb(255, 255, 255); box-sizing: border-box; color: rgb(0, 0, 0); display: inline; float: none; font-family: "Helvetica Neue",Helvetica,Arial,sans-serif; font-size: 12.6px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-t}, number={4}, publisher={Hacettepe University}