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Abstract
For 0 ? ? ? 1, Nikiforov proposed to study the spectral properties of the family of matrices A?(G) = ?D(G)+(1 ? ?)A(G) of a graph G, where D(G) is the degree diagonal matrix and A(G) is the adjacency matrix of G. The ?-spectral radius of G is the largest eigenvalue of A?(G). For a graph with two pendant paths at a vertex or at two adjacent vertices, we prove results concerning the behavior of the ?-spectral radius under relocation of a pendant edge in a pendant path. We give upper bounds for the ?-spectral radius for unicyclic graphs G with maximum degree ? ? 2, connected irregular graphs with given maximum degree and some other graph parameters, and graphs with given domination number, respectively. We determine the unique tree with the second largest ?-spectral radius among trees, and the unique tree with the largest ?-spectral radius among trees with given diameter. We also determine the unique graphs so that the difference between the maximum degree and the ?-spectral radius is maximum among trees, unicyclic graphs and non-bipartite graphs, respectively.
Highlights
For a graph with two pendant paths at a vertex or at two adjacent vertices, we prove two results concerning the behavior of the α-spectral radius under relocation of a pendant edge in a pendant path, which were conjectured in [27]
We show that the upper bound for the α-spectral radius of trees with maximum degree ∆ ≥ 2 in [26] holds for unicyclic graphs, and we give upper bounds for the α-spectral radius of connected irregular graphs with fixed maximum degree and some other graph parameters, and of graphs with fixed domination number, respectively
We determine the unique graphs so that the difference between the maximum degree and the αspectral radius is maximum among trees, unicyclic graphs and non-bipartite graphs, respectively
Summary
The spectral properties of the adjacency matrix and the signless Laplacian matrix of a graph have been investigated for a long time, see, e.g., [9,10]. Nikiforov [25] showed that the r-partite Turan graph is the unique graph with the largest α-spectral radius for. Determined the unique graph with the largest α-spectral radius among connected graphs on n vertices with diameter (at least) k. For a graph with two pendant paths at a vertex or at two adjacent vertices, we prove two results concerning the behavior of the α-spectral radius under relocation of a pendant edge in a pendant path, which were conjectured in [27]. We determine the unique graphs so that the difference between the maximum degree and the αspectral radius is maximum among trees, unicyclic graphs and non-bipartite graphs, respectively
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