Abstract
A unicyclic graph is a connected graph whose number of edges is equal to the number of vertices. Fan et al. (Discrete Math. 313:903-909, 2013) and Liu et al. (Electron. J. Linear Algebra 26:333-344, 2013) determined, independently, the unique unicyclic graph whose least Q-eigenvalue attains the minimum among all non-bipartite unicyclic graphs of order n with k pendant vertices. In this paper, we extend their results and determine the first three non-bipartite unicyclic graphs of order n with k pendant vertices ordering by least Q-eigenvalue.
Highlights
Let G = (V, E) be a simple undirected graph with vertex set V = V (G) = {v, v, . . . , vn} and edge set E = E(G), where n is called the order of G
We call the eigenvalues of Q(G) as the signless Laplacian eigenvalues or Q-eigenvalues of G
Fan et al [ ] and Liu et al [ ] determined, independently, the unique unicyclic graph whose least Q-eigenvalue attains the minimum among all non-bipartite unicyclic graphs of order n with k pendant vertices
Summary
We extend their results and determine the first three non-bipartite unicyclic graphs of order n with k pendant vertices ordering by least Q-eigenvalue. ([ ]) Let G be a connected non-bipartite graph of order n, and let x be an eigenvector of G corresponding to κ(G). ([ ]) Let G = G (v ) T(u) and G∗ = G (v ) T(u), where G is a non-bipartite connected graph containing two distinct vertices v , v , and T is a nontrivial tree. ([ ]) Let G be a non-bipartite connected graph of order n with diameter D. Unk(g), shown in Figure , denotes the unicyclic graph of order n with odd girth g and k pendant vertices, where g + l + k = n.
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