Graph theory is rapidly moving into the mainstream of mathematics mainly because of its applications in diverse fields. In this paper, we discuss certain ways of applying graph theoretical techniques to solve various problems and present the review of some of the applications.Mathematicians and scientists are becoming increasingly aware of the significance of graph theory as it is applied to other areas of science and being widely used to solve various real-world problems. Graph theory is used in organic chemistry, solid state physics and statistical mechanics, electrical engineering (communication networks and coding theory), optimization theory and operations research. The wide scope of these and other applications has been well documented [7 12]. The powerful combinatorial methods found in graph theory have also been used to prove fundamental in a variety of areas of pure mathematics. The best known of these methods are related to a part of graph theory called matchings, and the results from the area are used to prove Dilworth's chain decomposition theorem for finite partially ordered sets. An application of matchings shows that there is a common set of left and right coset representatives of a subgroup in a finite group. This result played an important role in Dharwardker's proof of the four-color theorem [16 18]. The existence of matchings in certain infinite bipartite graphs played an important role in Laczkovich's affirmative answer to Tarski's problem of whether a circle is piecewise congruent to a square. The proof of the existence of a subset of the real numbers R that is non-measurable in the Lebesgue sense is due to Theorem [15]. Surprisingly, this theorem can be proved using only discrete mathematics (bipartite graphs). There are many such examples of applications of graph theory to other parts of mathematics, but they remain scattered in the literature [11], for instance graph theoretical proof of Cantor-Schroder-Bernstein theorem [9], Fermats Little theorem [8], Neilson-Schrier theorem [4]. There are new and recent applications which include solving SNP problem and time tabling problem by using vertex cover algorithm [17]. There are many new and the older applications of travelling salesman problem(TSP). the TSP arises in many transportation and logistics applications. The new applications of TSP include genome sequencing, starlight interferometer program, scan chain optimization, DNA universal strings, whizzards 96 vehicle routing, a tour through MLB ballparks, touring airports, USA trip. References[1] P. Avery, Score sequences of oriented graphs, J. Graph Theory, Vol 15 (1991)251-257.[2] Ashay Dharwardker,A new algorithm for finding hamiltonian circuts (2004) http://www.geocities.com/dharwardker/hamilton.[3] Ashay Dharwardker,A new proof of the four color theorem (2000)

In this paper we extend the concept of energy to signed digraphs. We obtain Coulson's integral formula for energy of signed digraphs. Formulae for energies of signed directed cycles are computed and it is shown that energy of non cycle balanced signed directed cycles increases monotonically with respect to number of vertices. Characterization of signed digraphs having energy equal to zero is given. We extend the concept of non complete extended p sum (or briefly, NEPS) to signed digraphs. An infinite family of equienergetic signed digraphs is constructed. Moreover, we extend McClelland's inequality to signed digraphs and also obtain sharp upper bound for energy of signed digraph in terms the number of arcs. Some open problems are also given at the end.2010 Mathematics Subject Classification. 05C50, 05C22,05C76.

Abstract. For a simple connected graph G of order n, having Laplacian eigenvalues µ 1 , µ 2 , . . . , µ n−1 , µn = 0, the Laplacian-energy-like invariant (LEL) and the Kirchhoff index (Kf ) are defined as, respectively. In this paper, LEL and Kf are compared, and sufficient conditions for the inequality Kf (G) < LEL(G) are established.

The energy E(G) of a graph G is the sum of the absolute values of the eigenvalues of G. Bapat and Pati (Bull. Kerala Math. Assoc., 1 (2004), 129-132) proved that (a) E(G) is never an odd integer. We now show that (b) E(G) is never the square root of an odd integer. Furthermore, if r and s are integers such that r ≥ 1 and 0 ≤ s ≤ r − 1 and q is an odd integer, then E(G) cannot be of the form (2 s q) 1/r , a result that implies both (a) and (b) as special cases.

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