‎in this paper‎, ‎we define the common minimal dominating signed‎ ‎graph of a given signed graph and offer a structural‎ ‎characterization of common minimal dominating signed graphs‎. ‎in‎ ‎the sequel‎, ‎we also obtained switching equivalence‎ ‎characterizations‎: ‎$overline{s} sim cmd(s)$ and $cmd(s) sim‎ ‎n(s)$‎, ‎where $overline{s}$‎, ‎$cmd(s)$ and $n(s)$ are complementary‎ ‎signed gra...

let $p$ be a prime number and $n$ be a positive integer. the graph $g_p(n)$ is a graph with vertex set $[n]={1,2,ldots ,n}$, in which there is an arc from $u$ to $v$ if and only if $uneq v$ and $pnmid u+v$. in this paper it is shown that $g_p(n)$ is a perfect graph. in addition, an explicit formula for the chromatic number of such graph is given.

aim : to evaluate the role of  endoscopic ultrasonography (eus) in the diagnosis of gallbladder microlithiasis, sludge, and stone  in patients with clinical suspicion of cholecystitis, but with normal transabdominal ultrasonography (tus) during six months follow-up after laparosopic cholecystectomy (lct). background : endosonography has been shown to be highly sensitive in the detection of chol...

The degree set of a graph is the set of its degrees. Kapoor et al. [Degree sets for graphs, Fund. Math. 95 (1977) 189-194] proved that for every set of positive integers, there exists a graph of diameter at most two and radius one with that degree set. Furthermore, the minimum order of such a graph is determined. A graph is 2-self- centered if its radius and diameter are two. In this paper for ...

A combination of graph layouts in 3D space, interactive computer graphics, and virtual reality (VR) can increase the size of understandable networks for metabolic network visualization. Two models, the directed graph and the compound graph, were used to represent a metabolic network. The directed graph, or nonhierarchical visualization, considers the adjacency relationships. For the nonhierarch...

let $r$ be a commutative ring with identity. an ideal $i$ of a ring $r$is called an annihilating ideal if there exists $rin rsetminus {0}$ such that $ir=(0)$ and an ideal $i$ of$r$ is called an essential ideal if $i$ has non-zero intersectionwith every other non-zero ideal of $r$. thesum-annihilating essential ideal graph of $r$, denoted by $mathcal{ae}_r$, isa graph whose vertex set is the set...

the chromatic number of a graph g, denoted by χ(g), is the minimum number of colors such that g can be colored with these colors in such a way that no two adjacent vertices have the same color. a clique in a graph is a set of mutually adjacent vertices. the maximum size of a clique in a graph g is called the clique number of g. the turán graph tn(k) is a complete k-partite graph whose partition...

We describe a new animation technique for supporting interactive exploration of a graph, building on the wellknown radial tree layout method. When a node is selected to become the center of interest, the visualization performs an animated transition to a new layout. Our approach makes the transition easy to follow by linearly interpolating the polar coordinates of the nodes and enforcing constr...

The hierarchical edge bundle method clusters the edges of a radial layout to better understand and analyze graphs, but its effectiveness relies critically on the quality of the hierarchical organization of its nodes and edges. This paper proposes a novel graph visualization approach that extracts the community structure of a network and organizes it into a more balanced and meaningful hierarchy...