Computing the Sum of k Largest Laplacian Eigenvalues of Tricyclic Graphs

Abstract

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 ≤ (cid:98) g − 22 (cid:99) . 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 (cid:54) = 3 . We settle the conjecture under discussion also for tricyclic graphs having no pendant vertices when k = 3 .