Computing some role assignments of Cartesian Product of Graphs

In social networks, a role assignment is such that individuals play the same role, if they relate in the same way to other individuals playing counterpart roles. As a simple graph models a social network role assignment rises to the decision problem called r -Role Assignment whether it exists such an assignment of r distinct roles to the vertices of the graph. This problem is known to be NP-complete for any fixed r ≥ 2. The Cartesian product of graphs is one of the most studied operation on graphs and has numerous applications in diverse areas, such as Mathematics, Computer Science, Chemistry and Biology. In this paper, we determine the computational complexity of r -Role Assignment restricted to Cartesian product of graphs, for r = 2,3. In fact, we show that the Cartesian product of graphs is always 2-role assignable, however the problem of 3-Role Assignment is still NP-complete for this class.

Note on Reliability Analysis of Cartesian Product of Networks for Components

Connectivity is a critical parameter which can measure the reliability of networks. Let [Formula: see text] be a vertex set of [Formula: see text]. If [Formula: see text] has at least [Formula: see text] components, then [Formula: see text] is a [Formula: see text]-component cut of [Formula: see text]. The [Formula: see text]-component connectivity [Formula: see text] of [Formula: see text] is the vertex number of a smallest [Formula: see text]-component cut. Cartesian product of graphs is a useful method to construct a large network. We will use Cauchy–Schwarz inequality to determine the component connectivity of Cartesian product of some graphs.

Computing Bounds for Second Zagreb Coindex of Sum Graphs

Topological indices or coindices are one of the graph-theoretic tools which are widely used to study the different structural and chemical properties of the under study networks or graphs in the subject of computer science and chemistry, respectively. For these investigations, the operations of graphs always played an important role for the study of the complex networks under the various topological indices or coindices. In this paper, we determine bounds for the second Zagreb coindex of a well-known family of graphs called F -sum ( S -sum, R -sum, Q -sum, and T -sum) graphs in the form of Zagreb indices and coindices of their factor graphs, where these graphs are obtained by using four subdivision-related operations and Cartesian product of graphs. At the end, we illustrate the obtained results by providing the exact and bonded values of some specific F -sum graphs.

Study of Circular Distance in Graphs

Circular distance between vertices of a graph has a significant role, which is defined as summation of detour distance and geodesic distance. Attention is paid, this is metric on the set of all vertices of graph and it plays an important role in graph theory. Some bounds have been carried out for circular distance in terms of pendent vertices of graph . Some results and properties have been found for circular distance for some classes of graphs and applied this distance to Cartesian product of graphs〖 P〗_2×C_n. Including 〖 P〗_2×C_n, some graphs acted as a circular self-centered. Using this circular distance there exists some relations between various radii and diameters in path graphs. The possible applications were briefly discussed.

On very strongly perfect Cartesian product graphs

<abstract><p>Let $ G_1 \square G_2 $ be the Cartesian product of simple, connected and finite graphs $ G_1 $ and $ G_2 $. We give necessary and sufficient conditions for the Cartesian product of graphs to be very strongly perfect. Further, we introduce and characterize the co-strongly perfect graph. The very strongly perfect graph is implemented in the real-time application of a wireless sensor network to optimize the set of master nodes to communicate and control nodes placed in the field.</p></abstract>