Sharp upper bounds for the Laplacian graph eigenvalues Original Research Article

Let G=(V,E) be a simple connected graph and λ1(G) be the largest Laplacian eigenvalue of G. In this paper, we prove that: 1. λ1(G)=max{du+mu:uset membership, variantV} if and only if G is a regular bipartite or a semiregular bipartite graph, where du and mu denote the degree of u and the average of the degrees of the vertices adjacent to u, respectively. 2. image if and only if G is a regular bipartite graph or a semiregular bipartite graph, or a path with four vertices, where r=max{du+dv:uvset membership, variantE} and suppose xyset membership, variantE satisfies dx+dy=r, s=max{du+dv:uvset membership, variantE−{xy}}. 3. image if and only if G is a regular bipartite graph or a semiregular bipartite graph. 4. image with equality if and only if G is a regular bipartite graph or a semiregular bipartite graph, or a path with four vertices, where image and suppose xyset membership, variantE satisfies image image