Combinatorics, a branch of Mathematics that organizes and calculates elements under specific constraints, has implications in a variety of fields, including Computer Science, Physics, and Statistics [2, 5].The matroid serves as one of its essential concepts, since it abstracts the concept of independence in contexts such as Graphs, Networks, and Algebraic structures [15]. Matroids offer a connection between combinatorics and algebraic topology under their association with pure simplicial complexes, which are crucial for the analysis of topological spaces [13]. Hyperplane arrangement A (shortened to “arrangements”), which are collections of affine subspaces of codimension one in a vector space V over a field K, have a topology specified by invariants obtained combinatorically from their intersection lattice L(A) = {〖 ⋂ 〗_(H∈B)  H| B ⊆ A}  [8, 12]. Orlik and Solomon's construction of the Orlik-Solomon algebra A_* (A)  of an arrangement  A [7],calculates the cohomology H^* (M(A),K)  of the complement of the arrangement M(A)=V∖∪_(H∈A) H and presents techniques for studying cohomology.H^* (M(A),L(a)) with the rank one local system's coefficients L(a),a∈A_1 (A) [10, 16].It is possible to generalize H^* (M(A),K) to cohomology with local system L(a), if the non-vanishing cohomology H^* (A_* (A),a)) of the Orlik-Solomon algebra A_* (A)  exists [10, 14]. Our study integrates two important fields in mathematics with a wide range of applications, “Arrangements” and “Graphs,” by using matroids for transferring the structure used to construct the Orlik-Solomon algebra A_* (A_G) of the graphic arrangement A_G to reconstruct the Orlik-Solomon algebra A_* (G)  of a graph G using the duality between these two fields [8].Motivated by our goal, we used the bond lattice L(G) of a triangle-free graph G, to study the first non-vanishing cohomology H^* (A_* (G),a) of the Orlik-Solomon algebra A_* (G),where   a = a_1  -〖 a〗_t,1 < t ≤ l, and l is the number of edges in G. This paper splits into two sections: the first outlines the basic concepts, and the second investigates the cohomology of the Orlik-Solomon algebra for triangle-free graphs. We proved, if G has chordless 4-cycles, then H^2 (A_* (A_G), a) is the first non-vanished cohomology of A_* (A).