We estimate and characterize the edge congestion-sum measure for embeddings of various graphs such as cycles, wheels, and generalized wheels into arbitrary trees. All embedding algorithms apply an interesting general technique based on the consecutive label property. Our algorithms produce optimal values of sum of dilations and sum of edge-congestions in linear time.

The packing chromatic number χ ρ (G) of a graph G is the smallest integer k for which there exists a mapping Π : V (G) −→ {1, 2, ..., k} such that any two vertices of color i are at distance at least i + 1. It is a frequency assignment problem used in wireless networks, which is also called broadcasting coloring. It is proved that packing coloring is NP-complete for general graphs and even for trees. In this paper, we study the packing chromatic number of comb graph, circular ladder, windmill, H-graph and uniform theta graph.

ReO 3 lattices Silicate networksLet G(V , E) be a simple connected graph. A set S ⊆ V is a power dominating set (PDS) of G, if every vertex and every edge in the system is observed following the observation rules of power system monitoring. The minimum cardinality of a PDS of a graph G is the power domination number γ p (G). In this paper, we establish a fundamental result that would provide a lower bound for the power domination number of a graph. Further, we solve the power domination problem in polyphenylene dendrimers, Rhenium Trioxide (ReO 3 ) lattices and silicate networks.

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