As the Internet grows in size, it becomes crucial to understand how the speeds of links in the network must improve in order to sustain the pressure of new end-nodes being added each day. Although the speeds of links in the core and at the edges improve roughly according to Moore's law, this improvement alone might not be enough. Indeed, the structure of the Internet graph and routing in the network might necessitate much faster improvements in the speeds of key links in the network.In this paper, using a combination of analysis and extensive simulations, we show that the worst congestion in the Internet AS-level graph in fact scales poorly with the network size (n 1+Ω(1) , where n is the number of nodes), when shortestpath routing is used to route traffic between ASes. We also show, somewhat surprisingly, that policy-based routing does not exacerbate the maximum congestion when compared to shortest-path routing.Our results show that it is crucial to identify ways to alleviate this congestion to avoid some links from being perpetually congested. To this end, we show that the congestion scaling properties of Internet-like graphs can be improved dramatically by introducing moderate amounts of redundancy in the graph in terms of parallel edges between pairs of adjacent nodes.

As the Internet grows in size, it becomes crucial to understand how the speeds of links in the network must improve in order to sustain the pressure of new end-nodes being added each day. Although the speeds of links in the core and at the edges roughly improve according to Moore's law, this improvement alone might not be enough. Indeed, the structure of the Internet graph and routing in the network might necessitate much faster improvements in the speeds of key links in the network. In this paper, using a combination of analysis and extensive simulations, we show that the worst congestion in the Internet in fact scales poorly with the network size (n 1+Q^1 \ where n is the number of nodes), when shortest-path routing is used. We also show, somewhat surprisingly, that policy-based routing does not exacerbate the maximum congestion when compared to shortest-path routing. Our results show that it is crucial to identify ways to alleviate this congestion to avoid some links from being perpetually congested. To this end, we show that the congestion scaling properties of the Internet graph can be improved dramatically by introducing moderate amounts of redundancy in the graph in terms of parallel edges between pairs of adjacent nodes.

In the present paper, we introduce the classical mean labeling of graphs and investigate their related properties. Moreover, it is obtained that the line graph operation preserves the classical meanness property for some standard graphs.

In this study, we investigate a new kind of mean labeling of graph. The ladder graph plays an important role in the area of communication networks, coding theory, and transportation engineering. Also, we found interesting new results corresponding to classical mean labeling for some ladder-related graphs and corona of ladder graphs with suitable examples.

In this manuscript, we have examined the exponential mean labeling of some graphs obtained from duplicating operations.