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The set of trees with n vertices and the set of trees with perfect matchings are denoted by Fn and T2k , respectively. M. Hofmeister determined the .rst .ve maximum value of the largest eigenvalue of trees in Fn and gave the corresponding trees (Linear Algebra Appl. 260 (1997) 43–59). Focusing on the largest eigenvalue of trees in T2k , this paper will give(More)
OBJECTIVE Genetic variation in the serotonin-2C receptor encoded by the HTR2C gene is one of the genetic determinants of antipsychotic-induced weight gain. Peroxisome proliferator-activated receptors are nuclear receptors regulating the expression of genes involved in lipid and glucose metabolism. In this cross-sectional study, we investigated whether(More)
Let G = (V,E) be a graph without loops and multiple edges. Let n and m be the number of vertices and edges of G, respectively. Such a graph will be referred to as an (n,m)-graph. For v ∈ V (G), let d(v) be the degree of v. In this paper, we are concerned only with undirected simple graphs (loops and multiple edges are not allowed). Let G be a graph with n(More)
In this paper I am going to describe a way to calculate the number of spanning trees by arbitrary weight by an extension of Kirchhoff’s formula, also known as the matrix tree theorem. The ultimate goal is to describe an algorithm that calculates the number of minimal spanning trees of a graph on n vertices in O(M(n)), where M(n) is the time required to(More)
Inspired by 1 Ad1 am’s conjecture the isomorphism problem of circulant digraphs is widely investigated. In the literature, the spectrum method was to solve the isomorphism problem for the circulants of prime-power order by some people. In this paper, we develop the spectrum method to characterize the circulant digraphs of orders p and pq, where p and q are(More)
Let T + 2p be the set of all trees on 2p (p ≥ 1) vertices with perfect matchings. In this paper, we prove that for any tree T in T + 2p , the kth largest eigenvalue λk(T ) satisfies λk(T ) ≤ 1 2 “q ̊ p k ˇ − 1 + q ̊ p k ˇ + 3 ” (k = 1, 2, . . . , p). This upper bound is known to be best possible when k = 1. The set of trees obtained from a tree on p(More)
Let λ2(G) be the second smallest normalized Laplacian eigenvalue of a graph G. In this paper, we investigate the behavior on λ2(G) when the graph G is perturbed by separating an edge. This result can be used to determine all trees and unicyclic graphs with λ2(G) ≥ 1− √ 2 2 . Moreover, the trees and unicyclic graphswith λ2(G) = 1− √ 2 2 are also determined,(More)
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