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A graph G is supposed to be trivially perfect if, in each induced subgraph H of G, the number of maximal cliques in H equivalents to the size of a maximum independent set in H. Trivially perfect graphs is subclasses of notable perfect graphs and its characterization have numerous continuous applications and it is adequate to research its subclasses. Along with this idea, in this paper, it is discussed trivially perfect graphs on the windmill graph and demonstrated a few outcomes on trivially perfect 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>

Let L be a multiplicative lattice and M be a lattice module over L . In this paper, we assign a graph to M called residual division graph RG(M) in which the element N ∈ M is a vertex if there exists 0 M ≠ P ∈ M such that N P = 0 M and two vertices N 1 , N 2 are adjacent if N 1 N 2 = 0 M (where N 1 N 2 = N 1 : I M N 2 : I M I M ). It is proved that such a graph with the greatest element I M which does not belong to the vertex set is nonempty if and only if M is a prime lattice module. Also, we provide conditions such that R G M is isomorphic to a subgraph of Zariski topology graph Ģ X M with respect to X .

A graph G is said to be strongly perfect if every induced sub graph H has an independent set, meeting all maximal complete sub graphs of H. A strongly perfect graph is said to be very strongly perfect if it contains good independent set. It follows that every very strongly perfect graph is perfect not conversely. For example, the compliment of even cycle of length more than 4 is not very strongly perfect though it is perfect. In this paper, we characterise the classes of very strongly perfect graphs like cyclic graph, tadpole graph, barbell graph and friendship graph along with its independence number. 2010 AMS Classification: 05C17.