A λ harmonic graph G,a λ-Hgraph G for short,means that there exists a constant λ such that the equality λd(vi) =■holds for all i = 1,2,…,|V(G)|,where d(vi) denotes the degree of vertex vi.Let ni denote the number of vertices with degree i.This paper deals with the 3-Hgraphs and determines their degree series.Moreover,the 3-Hgraphs with bounded ni(1 ≤ i ≤ 7) are studied and some interesting results are obtained.
A λ harmonic graph G,a λ-Hgraph G for short,means that there exists a constant λ such that the equality λd(vi) =(vi,vj)∈E(G) d(vj) holds for all i = 1,2,…,|V(G)|,where d(vi) denotes the degree of vertex vi.In this paper,some harmonic properties of the complement and line graph are given,and some algebraic properties for the λ-Hgraphs are obtained.
Let G be a simple graph with n vertices and m edges.Let λ1,λ2,…,λn,be the adjacency spectrum of G,and let μ1,μ2,…,μn be the Laplacian spectrum of G.The energy n n of G is E(G) = ∑n i=1|λi|,while the Laplacian energy of G is defined as LE(G) =∑n i=1|μi - 2m |.n i=1 i=1 Let γ1,γ2,…,γn be the eigenvalues of Hermite matrix A.The energy of Hermite matrix as n HE(A) =∑n i=1 |γi - tr(A) | is defined and investigated in this paper.It is a natural generalization n i=1 of E(G) and LE(G).Thus all properties about energy in unity can be handled by HE(A).
