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Sharp upper bounds for the Laplacian graph eigenvalues

2002; Elsevier BV; Volume: 355; Issue: 1-3 Linguagem: Francês

10.1016/s0024-3795(02)00353-1

1873-1856

Yong-Liang Pan,

Metal-Organic Frameworks: Synthesis and Applications

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: λ1(G)=max{du+mu:u∈V} 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. λ1(G)=2+(r−2)(s−2) 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:uv∈E} and suppose xy∈E satisfies dx+dy=r, s=max{du+dv:uv∈E−{xy}}. λ1(G)=maxdu(du+mu)+dv(dv+mv)du+dv:uv∈E if and only if G is a regular bipartite graph or a semiregular bipartite graph. λ1(G)⩽2+(t−2)(b−2) with equality if and only if G is a regular bipartite graph or a semiregular bipartite graph, or a path with four vertices, wheret=maxdu(du+mu)+dv(dv+mv)du+dv:uv∈Eand suppose xy∈E satisfiesdx(dx+mx)+dy(dy+my)dx+dy=t,b=maxdu(du+mu)+dv(dv+mv)du+dv:uv∈E−{xy}.