Let G be a finite group. The character degree graph Gamma (G) of G is the graph whose vertices are the prime divisors of character degrees of G and two vertices p and q are joined by an edge if pq divides the character degree of G. Let L_n(q) be the projective special linear group of degree n over a finite field of order q. Khosravi et. al. have shown that the simple groups L_2(p^2), and L_2(p) where pin {7,8,11,13,17,19} are characterizable by the degree graphs and their orders. In this paper, we give a characterization of L_3(4) by using the character degree graph and its order.