ABSTRACT Several approaches for ordering graphs by spectral parameters are presented in the literature. We can find graph orderings either by the greatest eigenvalue (spectral radius or index) or by the sum of the absolute values of the... more
ABSTRACT Several approaches for ordering graphs by spectral parameters are presented in the literature. We can find graph orderings either by the greatest eigenvalue (spectral radius or index) or by the sum of the absolute values of the eigenvalues (the energy of a graph) or by the second smallest eigenvalue of the Laplacian matrix (the algebraic connectivity), among others. By considering the fact that the algebraic connectivity is related to the connectivity and shape of the graphs, several structural properties of graphs relative to this parameter have been studied. Hence, a large number of papers about ordering graphs by algebraic connectivity, mainly about trees and graphs with few cycles, have been published. This paper surveys the significant results concerning these topics, trying to focus on possible points to be investigated in order to understand the difficulties to obtain partial orderings via algebraic connectivity.
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A caterpillar is a tree in which the removal of all pendant vertices makes it a path. Let d ≥ 3 and n ≥ 6 be given. Let Pd−1 be the path of d − 1 vertices and Sp be the star of p + 1 vertices. Let p = [p1, p2, ..., pd−1] such that p1 ≥ 1,... more
A caterpillar is a tree in which the removal of all pendant vertices makes it a path. Let d ≥ 3 and n ≥ 6 be given. Let Pd−1 be the path of d − 1 vertices and Sp be the star of p + 1 vertices. Let p = [p1, p2, ..., pd−1] such that p1 ≥ 1, p2 ≥ 1, ..., pd−1 ≥ 1. Let C (p) be the caterpillar obtained from the stars Sp1 , Sp2 , ...,Spd−1 and the path Pd−1 by identifying the root of Spi with the i−vertex of Pd−1. Let n > 2 (d − 1) be given. Let C = {C (p) : p1 + p2 + ... + pd−1 = n − d + 1} and S = {C(p) ∈ C : pj = pd−j , j = 1, 2, · · · , ⌊ d − 1 2 ⌋}. In this paper, the caterpillars in C and in S having the maximum and the minimum algebraic connectivity are found. Moreover, the algebraic connectivity of a caterpillar in S as the smallest eigenvalue of a 2 × 2 - block tridiagonal matrix of order 2s × 2s if d = 2s + 1 or d = 2s + 2 is characterized. Work supported by CNPq 300563/94-9, Brazil.
