Computing the sum of k largest Laplacian eigenvalues of tricyclic graphs

Let G(V;E) be a simple graph with | V (G) | = n and | E(G) |= m. If S-k(G) is the sum of k largest Laplacian eigenvalues of G, then Brouwer's conjecture states that S-k(G) <= m + k(k+1)/2 for 1 <= k <= n. The girth of a graph G is the length of a smallest cycle in G. If g is the girth of G, then we show that the mentioned conjecture is true for 1 <= k <= left perpendicular g - 2 /2 right perpendicular. Wang et al. [Math. Comput. Model. 56 (2012) 60-68] proved that Brouwer's conjecture is true for bicyclic and tricyclic graphs whenever 1 <= k <= n with k not equal 3. We settle the conjecture under discussion also for tricyclic graphs having no pendant vertices when k = 3