let $g$ be a connected graph of order $3$ or more and $c:e(g)rightarrowmathbb{z}_k$‎ ‎($kge 2$) a $k$-edge coloring of $g$ where adjacent edges may be colored the same‎. ‎the color sum $s(v)$ of a vertex $v$ of $g$ is the sum in $mathbb{z}_k$ of the colors of the edges incident with $v.$ the $k$-edge coloring $c$ is a modular $k$-edge coloring of $g$ if $s(u)ne s(v)$ in $mathbb{z}_k$ for all pa...

In this paper, we investigate circular chromatic number of Mycielski construction of graphs. It was shown in [20] that t Mycielskian of the Kneser graph KG(m,n) has the same circular chromatic number and chromatic number provided that m + t is an even integer. We prove that if m is large enough, then χ(M (KG(m,n))) = χc(M (KG(m,n))) where M t is t Mycielskian. Also, we consider the generalized ...

Let G be a finite connected simple graph and μ(G) be the Mycielskian of G. We show that for connected graphs G and H, μ(G) is isomorphic to μ(H) if and only if G is isomorphic to H. Furthermore, we determine the energy of the Mycielskian of a connected regular graph G in terms of the energy E(G) of G, where the energy of G is the sum of the absolute values of the eigenvalues of G. The energy of...

For a (molecular) graph, the hyper Zagreb index is defined as HM(G) = ∑ uv∈E(G) (dG(u) + dG(v)) 2 and the hyper Zagreb coindex is defined asHM(G) = ∑ uv/ ∈E(G) (dG(u)+dG(v)) 2. In this paper, the hyper Zagreb indices and its coindices of edge corona product graph, double graph and Mycielskian graph are obtained.

Since the characterization of Gromov hyperbolic graphs seems a too ambitious task, there are many papers studying the hyperbolicity of several classes of graphs. In this paper, it is proven that every Mycielskian graph GM is hyperbolic and that δ(GM) is comparable to diam(GM). Furthermore, we study the extremal problems of finding the smallest and largest hyperbolicity constants of such graphs;...

Let ω(G) and χ(G) be the clique number and the chromatic number of a graph G. Mycielski [11] presented a construction that for any n creates a graph Mn which is triangle-free (ω(G) = 2) with χ(G) > n. The starting point is the complete graph of two vertices (K2). M(n+1) is obtained from Mn through the operation μ(G) called the Mycielskian of a graph G. We first define the operation μ(G) and the...