Browsing by Author "Alikhani, Saeid"

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Access status: Open Access ,
Dominating sets and domination polynomials of certain graphs, II
(2010) Alikhani, Saeid; Peng, Yee Hock
The domination polynomial of a graph $G$ of order $n$ is the polynomial $D(G,x) = \sum _{i=\gamma(G)}^n d(G,i)x^i$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$, and $\gamma (G)$ is the domination number of $G$. In this paper, we obtain some properties of the coefficients of $D(G,x)$. Also, by study of the dominating sets and the domination polynomials of specific graphs denoted by $G^{\prime}(m)$, we obtain a relationship between the domination polynomial of graphs containing an induced path of length at least three, and the domination polynomial of related graphs obtained by replacing the path by shorter path. As examples of graphs $G^{\prime}(m)$, we study the dominating sets and domination polynomials of cycles and generalized theta graphs. Finally, we show that, if $n \equiv 0,2(mod\, 3)$ and $D(G,x) = D(C_n, x)$, then $G = C_n$.
Access status: Open Access ,
Introduction to dominated edge chromatic number of a graph
(Wydawnictwa AGH, 2021) Piri, Mohammad R.; Alikhani, Saeid
We introduce and study the dominated edge coloring of a graph. A dominated edge coloring of a graph $G$, is a proper edge coloring of $G$ such that each color class is dominated by at least one edge of $G$. The minimum number of colors among all dominated edge coloring is called the dominated edge chromatic number, denoted by $\chi_{dom}^{\prime}(G)$. We obtain some properties of $\chi_{dom}^{\prime}(G)$ and compute it for specific graphs. Also examine the effects on $\chi_{dom}^{\prime}(G)$, when $G$ is modified by operations on vertex and edge of $G$. Finally, we consider the $k$-subdivision of $G$ and study the dominated edge chromatic number of these kind of graphs.