1. CMB 2014 (vol 57 pp. 573)
|Some Results on the Domination Number of a Zero-divisor Graph|
In this paper, we investigate the domination, total domination and semi-total domination numbers of a zero-divisor graph of a commutative Noetherian ring. Also, some relations between the domination numbers of $\Gamma(R/I)$ and $\Gamma_I(R)$, and the domination numbers of $\Gamma(R)$ and $\Gamma(R[x,\alpha,\delta])$, where $R[x,\alpha,\delta]$ is the Ore extension of $R$, are studied.
Keywords:zero-divisor graph, domination number
2. CMB 2014 (vol 57 pp. 520)
|Maximizing the Index of Trees with Given Domination Number|
The index of a graph $G$ is the maximum eigenvalue of its adjacency matrix $A(G)$. In this paper we characterize the extremal tree with given domination number that attains the maximum index.
Keywords:trees, spectral radius, index, domination number
3. CMB 2013 (vol 57 pp. 141)
|Size, Order, and Connected Domination|
We give a sharp upper bound on the size of a triangle-free graph of a given order and connected domination. Our bound, apart from strengthening an old classical theorem of Mantel and of TurÃ¡n , improves on a theorem of Sanchis. Further, as corollaries, we settle a long standing conjecture of Graffiti on the leaf number and local independence for triangle-free graphs and answer a question of Griggs, Kleitman and Shastri on a lower bound of the leaf number in triangle-free graphs.
Keywords:size, connected domination, local independence number, leaf number
4. CMB 2011 (vol 56 pp. 407)
|On Domination in Zero-Divisor Graphs|
We first determine the domination number for the zero-divisor graph of the product of two commutative rings with $1$. We then calculate the domination number for the zero-divisor graph of any commutative artinian ring. Finally, we extend some of the results to non-commutative rings in which an element is a left zero-divisor if and only if it is a right zero-divisor.
Keywords:zero-divisor graph, domination