The undirected power graph (or simply graph) of a group $G$, denoted by $P(G)$, is whose vertices are the elements in which two $u$ and $v$ connected an edge between if only either $u=v^i$ or $v=u^j$ for some $i$, $j$. A number important classes, including perfect graphs, cographs, chordal split threshold can be defined structurally terms forbidden induced subgraphs. We examine each these five ...

The strong power graph G(G) of a group G with elements {1, 2, . . . , n} is a simple graph whose vertices being the elements of G and two distinct vertices i and j are adjacent in G(G) if im1 = jm2 for some positive integers m1,m2 < n. In this paper atfirst we define the strong power graphs and then study some of its properties. Further, we characterise power graphs and strong power graphs and ...

The power graph of a group G is the graph whose vertex set is G and two distinct vertices are adjacent if one is a power of the other. In this paper, the minimum degree of power graphs of certain classes of cyclic groups, abelian p-groups, dihedral groups and dicyclic groups are obtained. It is ascertained that the edge-connectivity and minimum degree of power graphs are equal, and consequently...

emph{The (undirected) power graph on the conjugacy classes} $mathcal{P_C}(G)$ of a group $G$ is a simple graph in which the vertices are the conjugacy classes of $G$ and two distinct vertices $C$ and $C'$ are adjacent in $mathcal{P_C}(G)$ if one is a subset of a power of the other. In this paper, we describe groups whose associated graphs are $k$-regular for $k=5,6$.

Large graphs abound around us – online social networks, Web graphs, the Internet, citation networks, protein interaction networks, telephone call graphs, peer-to-peer overlay networks, electric power grid networks, etc. Many reallife graphs are power-law graphs. A fundamental challenge in today’s Big Data world is storage and processing of these large-scale power-law graphs. In this thesis, we ...

In this paper, we introduce the concept of k-power domination which is a common generalization of domination and power domination. We extend several known results for power domination to k-power domination. Concerning the complexity of the k-power domination problem, we first show that deciding whether a graph admits a k-power dominating set of size at most t is NP-complete for chordal graphs a...

For a graph $G = (V, E)$, a partition $pi = {V_1,$ $V_2,$ $ldots,$ $V_k}$ of the vertex set $V$ is an textit{upper domatic partition} if $V_i$ dominates $V_j$ or $V_j$ dominates $V_i$ or both for every $V_i, V_j in pi$, whenever $i neq j$. The textit{upper domatic number} $D(G)$ is the maximum order of an upper domatic partition. We study the properties of upper domatic number and propose an up...