Background and Objectives: With the great growth of applications sensitive to latency, and efforts to reduce latency and cost and to improve the quality of service on the Internet of Things ecosystem, cloud computing and communication between things and the cloud are costly and inefficient; Therefore, fog computing has been proposed to prevent sending large volumes of data generated by things to cloud centers and, if possible, to process some requests. Today's advances in 5G networks and the Internet of Things show the benefits of fog computing more than ever before, so that services can be delivered with very little delay as resources and features of fog nodes approach the end user.Methods: Since the cloud-fog paradigm is a layered architecture, to reduce the overall delay, the fog layer is divided into two sub-layers in this paper, including super nodes and ordinary nodes in order to use the coverage of super peer networks to use the connections between fog nodes in addition to taking advantage of the features of that network and improving the performance of large-scale systems. It causes fog nodes to interact with each other in processing requests and fewer data will be sent to the cloud, resulting in a reduction in overall latency. To reduce the cost of bandwidth used among fog nodes, we have organized a sub-layer of super nodes in the form of a Perfect Difference Graph (PDG). The new platform proposed for aggregation of fog computing and Internet of Things (FOT) is called the P2P-based Fog supported Platform (PFP).Results: We evaluate the utility of our proposed method by applying ifogsim simulator and the results achieved are as follows: (1) power consumption parameter in our proposed method 24% and 38% have improved compared to the structure three-layer fog computing architecture and without fog layer respectively; (2) network usage parameter in our proposed method 26% and 32% have improved compared to the structure three-layer fog computing architecture and without fog layer respectively; (3) average response time parameter in our proposed method 17% and 58% have improved compared to the structure three-layer fog computing architecture and without fog layer respectively; and (4) delay parameter in our proposed method 1% and 0.4% have improved compared to the structure three-layer fog computing architecture and without fog layer respectively.Conclusion: Numerical results obtained from the simulation show that the delay and cost parameters are significantly improved compared to the structure without fog layer and three-layer fog computing architecture. Also, the results show that increasing number of things has the same effect in all cases.

In this paper, we study Perfect 2-coloring of the quartic Graphs with at most 8 vertices. The problem of the existence of Perfect coloring is a generalization of the concept of completely regular codes, given by Delsarte.

Let $G$ be a group and $S$ be the collection of all non-trivial proper subgroups of $G$. The co-maximal subgroup Graph $\Gamma(G)$ of a group $G$ is defined to be a Graph with $S$ as the set of vertices and two distinct vertices $H$ and $K$ are adjacent if and only if $HK=G$. In this paper, we study the comaximal subgroup Graph on finite dihedral groups. In particular, we study order, maximum and minimum degree, diameter, girth, domination number, chromatic number and Perfectness of comaximal subgroup Graph of dihedral groups. Moreover, we prove some isomorphism results on comaximal subgroup Graph of dihedral groups.

The annihilating-ideal Graph of a commutative ring R with unity is de-, ned as the Graph AG(R) whose vertex set is the set of all non-zero ideals with non-zero annihilators and two distinct vertices I and J are adjacent if and only if IJ = 0. Nikan-dish et. al. proved that AG(Zn) is weakly Perfect. In this short paper, we characterize n for which AG(Zn) is Perfect.

Let‎ ‎$G =(V_G‎ , ‎E_G)$‎ ‎be a Graph and let $I$ be a finite set of size $m\geq 1$‎. ‎A mapping $T:V_G \rightarrow I$‎ ‎is called a Perfect $m$-coloring with a parameter matrix $A = (a_{ij})_{i,j\in I }$‎ ‎of $G$ if it is surjective and for all $i,j$‎, ‎every vertex of color $i$ has $a_{i j}$ neighbors of color $j$‎. ‎In this paper‎, ‎we classify all the realizable parameter matrices of Perfect 3-colorings of the line Graphs of the connected bicubic Graphs of order at most 12‎.

In this paper we give a characterization of unmixed tripartite Graphs under certain conditions which is a generalization of a result of Villarreal on bipartite Graphs. For bipartite Graphs two different characterizations were given by Ravindra and Villarreal. We show that these two characterizations imply each other.