**ABSTRACT** – Applications of graph theory are primarily, but not exclusively, concerned with labeled graphs and various specializations of these.Structures that can be represented as graphs are ubiquitous, and many problems of practical interest can be represented by graphs. The link structure of a relationship between a boy and a girl could be represented by a directed graph: the vertices are the boys and girls having some degree of inclination for each other, available for an open or closed system and a directed edge from a boy B[i] to a girl G[j] exists if and only if both B[i] and G[i] have an equal or almost equal inclination toward each other.

In the mathematical discipline of graph theory, a matching or independent edge set in a graph is a set of edges without common vertices. It may also be an entire graph consisting of edges without common vertices. Given a graph G = (B[i],G[j] for all 0

A vertex(a Boy or a Girl) is matched (or saturated) if it is incident to an edge in the matching. Otherwise the vertex is unmatched.A maximal matching is a matching M of a graph G with the property that if any edge not in M is added to M, it is no longer a matching and the system becomes unstable. In other words, a matching M of a graph G is maximal if every edge in G has a non-empty intersection with at least one edge in M. The whole system has got maximum stability when each vertex(Boys and Girls) are part of one and only one Matching M, with no intersection at all with other Matchings. In simpler words, if each Boy B[i] has an inclination for only one Girl G[j] and vice-versa. For maximum stability, i should be equal to j, for all i,j=Z+.

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**PROPOSAL – **Modern Game Theory has always been used to predict the stability of a system having multiple forces, taking into account the behaviour of other points in the system. A Simple Boy Girl Relationship can be expressed as a directed graph, with each vertex representing a Boy and a Girl, inclined towards each other. Such a Directed Edge, in the absence of any other third isolated node or any other unmatched graph, is the most stable system possible.

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**AXIOM 1 – **A single Node(A single Boy B[i], or a Single Girl G[j]), is stable for the time interval in which it is isolated. In the presence of other single nodes, such an isolated node will eventually form a directed graph with some other node, which may be isolated or part of a matching graph.

**AXIOM 2 – **As stated earlier, a directed graph consisting of only two nodes, one of which is a boy B1, and the other girl G1, with equal inclination towards each other, is the most stable system possible. All complex systems whether stable or unstable, strive to attain this most stable system by various means.

**Corollary –** If this directed edge between B1 and G1 is a part of a set of B[i] and G[j] as shown below, and B1 and G1 do not have equal inclination towards each other, then there is a possibility of B1 and G1 forming the maximum stable system with rest of G[j] and B[i] respectively. In case of even number of total nodes, the whole system still remains stable because of smaller stable sub-systems. In case of odd number of total nodes, we have a highly unstable super-system with constantly shifting stable sub-systems and one always isolated node.

**AXIOM 3 –** A system with two B[i]s and one G[j] or two G[j]s and one B[i], is a highly unstable system, also known as the Classic L Triangle. The system constantly tries to attain stability by B[i]-G[j] pairing, but there always exists a single unstable B[i] or G[j] at any point of time, resulting in an overall unstable system.

**THEOREM : **The Degree of instability of a system, is defined as the number of isolated B[i] or G[j] nodes , divided by the total number of directed graphs in the system. Consequently, a system with higher U-score tends to be more stable than a system with lower **U-score**.

The U-score for fully stable system is Zero.

The U-score for the classic L-triangle is 1.

**AXIOM 4 **– For systems with 3 or more nodes, as stated earlier, the system tends to form stable sub-systems consisting of directed graphs between B[i]s and G[j]s. The U-score of the system decreases with the number of directed graphs in the system and increases with the number of isolated nodes in the overall system.

A system consisting of only G[j]s or only B[i]s , has a U-score of infinity and is highly unstable.

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**CONCLUSION: **Thus we see how Graph theory can be applied for Modern Day relationships to get some logical analysis. The systems considered above are but ideal, and the following assumptions have been made, which may deviate from a real case scenario.

1) Each B[i] and G[j] is assumed to have the same inclinations towards all other G[j]s and B[i]s. In Reality, the L-Bond between no two B[i]-G[j] pair is equal in strength,

2) It has been found after extensive study, that B[i]s and G[j]s also interact among themselves and have a tendency to form directed graphs amongst themselves, often in the absence of nodes of the other kind. However for the above study, such interactions have not been considered.

3) As per Classical Game theory, each B[i] and each G[j] has multiple choices at any point of time. The overall stability of the system, ultimately depends on the choices made by these nodes. Hence a detailed study of the application of game theory also needs to be considered for the above problem.

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With these assumptions in place, it can be safely said, that Mathematics can be used with proper care, to explain something as random and chaotic as Modern Day Relationships. As it’s always said, “*There is always some madness in love. But there is also always some reason in madness.”*

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