This paper on the graph theory is aiming at presenting what is considered to be the basic material, combined with broad variety of applications, to other branches of mathematics and also to the real world problems. An overview of two applications of graph theory are tackled within the scope of this paper in the context of computer science professionalism.

**Relevance of graph theory to ad-hoc networks**

This covers the discussion of the role the graph theory play in relation to the issues pertaining to the Mobile Adhoc Networks (MANETS). The Adhoc networks consider issues as routing, scalability, modelling the network, connectivity and simulation (Jajodia, 2013). The capability of the network being modeled as a graph, the analysis of these issues can be done through the model. Graphs are algebraically represented in form of matrices. Networks too can be automated through algorithms. Issues involving nodes mobility, node density, packet routing and link formations between the nodes have to be simulated. The simulation of these concepts uses the random graph theory. The issues of connectivity are analyzed by use of graph spanners. For instance, the initial introduction of a k-spanner graph or a k-spanner or a geometric spanner as a weighted graph over some sets of points as vertices with every pair of vertices being a path between them having the weight at most k times, and the spatial distance between the points, for a fixed value of k. another way of analyzing the connectivity issues is through sparsification, proximity graphs, and spectral graph theory. The proximity graphs are generally graphs in which the connection of the two vertices is done by an edge if and only if thevertices satisfy specific geometric requirements. There are also numerous algorithms available for the analysis of the congestion in MANETâ€™swhere the modeling of these graphs is done on the basis of graph theoretical ideas.

**Computer Network Security**

The second application of the graph theory within the context of computer science is the algorithm of the graphs in computer network security. The vertex over algorithm recently have been used to slate the transmission of stealth worms on the large computer networks thereby designing optimal strategies to protect the network from such types of virus attacks in real-time.

With respect to the conference by SOFSEM & Bielikova (2012), this simulation is carried out in a huge internet-like virtual network and resulted that the combinatorial topology of routing can have a big impact on the propagation of the worm and therefore some servers play a more significant and essential role than others. The real-time capability of worm identification is significant in hindering worm propagation. The whole idea revolves around finding a minimum vertex cover in the graph, that is, the vertices which are the routing servers and whose edges become the possible dynamic connections of the routing servers. This give an optimal solution to worm propagation and for the network defense strategy design.

**Advancement of graph theory in computer science**

Graphical representation of information tend to be among the ubiquitous models of the human-made and natural structures. The graphs can be used in the modeling of several process and relations dynamics in computer science, and also across all the systems. The graph theory has advanced in the computer science by its inclusion in the graphical representations using the computer systems in connection to graph theoretical data structures, for instance, the matrix and list structures.

### Web documents clustering by using graph model

The enhanced representation of the web documents through clustering them use graphs rather vectors. This has been advanced in that there is use of classical k-meaning clustering algorithm by use of maximum common sub-graph measure of distance rather than the normal distance measure and the aspects of median graphs but not centroid calculations.

**Modeling sensor networks in graphs**

The modeling of sensor networks in form of graphs has been basically to analyze the efficiency in communications. The sensor networks are applied in collection of environmental data, tracking mobile objects, defense applications among other varied applications. There is conversion of a message pruning tree incurring the minimum cost to track the moving objects in a wireless sensor network (Bondy & Murty, 2002). The Voronoi graph have been taken in the modeling of the sensor network. The Oronoi diagram is a special type of decomposition of a metric space-distance of a particular discrete set of objects within the space.

A number of challenges of interest with respect to computer science can be phrased in form of graph problems thereby resulting into application of game theory.

**Measurement.**

The measurement-based quantum computation is a modeling computation that do not have a counterpart in real life. The computation in this model is done by making measurements on special category of quantum states (Jajodia, 2013). The states are graph states, since each can be uniquely associated with undirected graph having a number of vertices that are equal to the qubitsâ€™ number in the graph state.

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