On the Randić energy of caterpillar graphs

Revista Integración

Caterpillar trees, or simply Caterpillar, are trees such that when we remove all their leaves (or end edge) we obtain a path. The number of nonisomorphic caterpillars with n ≥ 2 edges is 2n−3 + 2⌊(n−3)/2⌋. Using a new sum of graphs, introduced in this paper, we provided a new proof of this result.

The electronic journal of linear algebra ELA

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.

The energy E(G) of a graph G is the sum of the absolute values of the eigenvalues of G of its adjacency matrix. The Laplacian energy LE(G) of a graph G is the sum of absolute values of its Laplacian eigenvalues. In this paper, we provide a MATLAB program, to calculate the energy and Laplacian energy of certain planar graphs namely n-regular caterpillar and Necklace.

2015

Let G be a simple graph of order n. The energy E(G) of G is the sum of the absolute values of the eigenvalues of G. The Randi´cRandi´c matrix of G, denoted by R(G), is defined as the n×n matrix whose (i, j)-entry is (didj) −1 2 if vi and vj are adjacent and 0 for another cases. The Randi´cRandi´c energy RE(G) of G is the sum of absolute values of the eigen-values of R(G). In this paper we compute the energy and the Randi´cRandi´c energy for certain graphs. We also propose a conjecture on the Randi´cRandi´c energy.

Linear Algebra and its Applications, 2010

2016

Let G be a simple graph with vertex set V (G) = {v 1 , v 2 ,. .. , v n }. The Randić matrix of G, denoted by R(G), is defined as the n × n matrix whose (i, j)-entry is (d i d j) −1 2 if v i and v j are adjacent and 0 for another cases. Let the eigenvalues of the Randić matrix R(G) be ρ 1 ≥ ρ 2 ≥. .. ≥ ρ n which are the roots of the Randić characteristic polynomial n i=1 (ρ − ρ i). The Randić energy RE of G is the sum of absolute values of the eigenvalues of R(G). In this paper we compute the Randić characteristic polynomial and the Randić energy for specific graphs G.

Electronic Notes in Discrete Mathematics, 2009

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 P d−1 be the path on d − 1 vertices and K 1,p be the star of p + 1 vertices. Let p = [p 1 , p 2 , ..., p d−1 ] such that ∀i, 1 ≤ i ≤ d − 1, p i ≥ 1. Let C (p) be the caterpillar obtained from d − 1 stars K 1,p i and the path P d−1 by identifying the root of K 1,p i with the i−vertex of P d−1 . For a given n ≥ 2 (d − 1),

The characteristic polynomials of the adjacency matrix of line graphs of caterpillars and then the characteristic polynomials of their Laplacian or signless Laplacian matrices are characterized, using recursive formulas. Furthermore, the obtained results are applied on the determination of upper and lower bounds on the algebraic connectivity of these graphs.

Discrete Applied Mathematics, 2018

Let G be a simple undirected graph. A broadcast on G is a function f : V (G) → N such that f (v) ≤ e G (v) holds for every vertex v of G, where e G (v) denotes the eccentricity of v in G, that is, the maximum distance from v to any other vertex of G. The cost of f is the value cost(f) = v∈V (G) f (v). A broadcast f on G is independent if for every two distinct vertices u and v in G, d G (u, v) > max{f (u), f (v)}, where d G (u, v) denotes the distance between u and v in G. The broadcast independence number of G is then defined as the maximum cost of an independent broadcast on G. In this paper, we study independent broadcasts of caterpillars and give an explicit formula for the broadcast independence number of caterpillars having no pair of adjacent trunks, a trunk being an internal spine vertex with degree 2.

Linear Algebra and its Applications, 2007

The higher Randić index R t (G) of a simple graph G is defined as R t (G) = i 1 i 2 •••i t+1 1 δ i 1 δ i 2 • • • δ i t+1 , where δ i denotes the degree of the vertex i and i 1 i 2 • • • i t+1 runs over all paths of length t in G. In [J.A. Rodríguez, A spectral approach to the Randić index, Linear Algebra Appl. 400 (2005) 339-344], the lower and upper bound on R 1 (G) was determined in terms of a kind of Laplacian spectra, and the lower and upper bound on R 2 (G) were done in terms of kinds of adjacency and Laplacian spectra. In this paper we characterize the graphs which achieve the upper or lower bounds of R 1 (G) and R 2 (G), respectively.