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Let G(V, E) be a unicyclic graph, Cm be a cycle of length m and Cm G, and ui ∈ V(Cm). The G - E(Cm) are m trees, denoted by Ti, i = 1, 2,..., m. For i = 1, 2,..., m, let eui be the excentricity of ui in Ti and ec = max{eui : i = 1, 2 , m}. Let κ = ec+1. Forj = 1,2,...,k- 1, let δij = max{dv : dist(v, ui) = j,v ∈ Ti}, δj = max{δij : i = 1, 2,..., m}, δ0 = max{dui : ui ∈ V(Cm)}. Then λ1(G)≤max{max 2≤j≤k-2 (√δj-1-1+√δj-1),2+√δ0-2,√δ0-2+√δ1-1}. If G ≌ Cn, then the equality holds, where λ1 (G) is the largest eigenvalue of the adjacency matrix of G. 相似文献
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The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Let U(k) be the set of all unicyclic graphs with a perfect matching. Let C
g(G) be the unique cycle of G with length g(G), and M(G) be a perfect matching of G. Let U
0(k) be the subset of U(k) such that g(G)≡ 0 (mod 4), there are just g/2 independence edges of M(G) in C
g(G) and there are some edges of E(G)\ M(G) in G\ C
g(G) for any G∈U
0(k). In this paper, we discuss the graphs with minimal and second minimal energies in U
*(k) = U(k)\ U
0(k), the graph with minimal energy in U
0(k), and propose a conjecture on the graph with minimal energy in U(k).
相似文献
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For a simple graph G, the energy E(G) is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let Undenote the set of all connected unicyclic graphs with order n, and Ur n= {G ∈ Un| d(x) = r for any vertex x ∈ V(Cl)}, where r ≥ 2 and Cl is the unique cycle in G. Every unicyclic graph in Ur nis said to be a cycle-r-regular graph.In this paper, we completely characterize that C39(2, 2, 2) ο Sn-8is the unique graph having minimal energy in U4 n. Moreover, the graph with minimal energy is uniquely determined in Ur nfor r = 3, 4. 相似文献
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图G的原子键连通性指数的定义如下:ABC(G)=∑uv∈E(G)((du+dv-2)dudv)(1/2).其中du、dv分别表示图G的边uv的2个端点u、v的度数.ABC指数已被证实为研究烷烃的稳定性以及环烷烃的应变能提供了一个很好的模型.讨论了n阶化学单圈图,给出了其ABC指数的可达的下界及其相应的极图 相似文献
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Let G =(V, E) be a simple connected graph with n(n ≥ 3) vertices and m edges,with vertex degree sequence {d1, d2,..., dn}. The augmented Zagreb index is defined as AZI =AZI(G)=∑ij∈E(didj/di+dj-2)3. Using the properties of inequality, we investigate the bounds of AZI for connected graphs, in particular unicyclic graphs in this paper, some useful conclusions are obtained. 相似文献
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LARGEST EIGENVALUE OF A UNICYCLIC MIXED GRAPH 总被引:3,自引:0,他引:3
FanYizheng 《高校应用数学学报(英文版)》2004,19(2):140-148
The graphs which maximize and minimize respectively the largest eigenvalue over all unicyclic mixed graphs U on n vertices are determined. The unicyclic mixed graphs U with the largest eigenvalue λ1 (U)=n or λ1 (U)∈ (n ,n 1] are characterized. 相似文献
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k圈图是边数等于顶点数加k-1的简单连通图.文中确定了不含三圈的k圈图的拟拉普拉斯谱半径的上界,并刻画了达到该上界的极图.此外,文中确定了拟拉普拉斯谱半径排在前五位的不含三圈的单圈图,排在前八位的不含三圈的双圈图.最后说明文中所得结论对不含三圈的k圈图的拉普拉斯谱半径也成立. 相似文献
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