Introduction/purpose: The Laplacian energy (LE) is the sum of absolute values of the terms μi-2m/n, where μi, i=1,2,…,n, are the eigenvalues of the Laplacian matrix of the graph G with n vertices and ...m edges. In 2006, another quantity Z was introduced, based on Laplacian eigenvalues, which was also named „Laplacian energy“. Z is the sum of squares of Laplacian eigenvalues. The aim of this work is to establish relations between LE and Z. Results: Lower and upper bounds for LE are deduced, in terms of Z. Conclusion: The paper contributes to the Laplacian spectral theory and the theory of graph energies. It is shown that, as a rough approximation, LE is proportional to the tem (Z-4m2/n)1/2. / Введение/цель: Энергия Лапласа (LE) представляет собой сумму абсолютных значений μi-2m/n, где μi, i=1,2,...,n являются собственными значениями G графы матрицы Лапласа с вершинами n и ребром m. В 2006 году была введена величина Z, основанная на характерных значениях Лапласа, которая получила название «Лапласова энергия». Z – это сумма квадратов собственных значений Лапласа. Целью данной работы является установление соотношений между LE и Z. Результаты: Нижняя и верхняя границы для LE выводятся из функции Z. Выводы: Статья способствует спектральной теории Лапласа и теории энергии графов. В грубой аппроксимации было показано, что LE пропорциональна (Z-4m2/n)1/2. / Uvod/cilj: Laplasova energija (LE) jeste suma apsolutnih vrednosti pojmova mi -2m/n, gde su mi , i=1,2,...,n, sopstvene vrednosti Laplasove matrice grafa G sa n vrhova i m ivica. Godine 2006. uvedena je druga veličina Z, zasnovana na Laplasovim svojstvenim vrednostima, koja je takođe nazvana "Laplasova energija". Z je suma kvadrata Laplasovih svojstvenih vrednosti. Cilj ovog rada je nalaženje odnosa između LE i Z. Rezultati: Donja i gornja granica za LE određene su kao funkcije od Z. Zaključak: Rad doprinosi Laplasovoj spektralnoj teoriji i teoriji energije grafova. Pokazano je da je, kao gruba aproksimacija, LE proporcionalna sa (Z-4m2 /n)1/2.
More on the Laplacian Estrada index Zhou, Bo; Gutman, Ivan
Applicable analysis and discrete mathematics,
10/2009, Volume:
3, Issue:
2
Journal Article
Peer reviewed
Open access
Let G be a graph with n vertices and let ?1, ?2, . . . , ?n be its Laplacian eigenvalues. In some recent works a quantity called Laplacian Estrada index was considered, defined as LEE(G)?n1 e?i. We ...now establish some further properties of LEE, mainly upper and lower bounds in terms of the number of vertices, number of edges, and the first Zagreb index.
nema
On incidence energy of graphs Das, Kinkar Ch; Gutman, Ivan
Linear algebra and its applications,
04/2014, Volume:
446
Journal Article
Peer reviewed
Open access
Let G=(V,E) be a simple graph with vertex set V={v1,v2,…,vn} and edge set E={e1,e2,…,em}. The incidence matrix I(G) of G is the n×m matrix whose (i,j)-entry is 1 if vi is incident to ej and 0 ...otherwise. The incidence energy IE of G is the sum of the singular values of I(G). In this paper we give lower and upper bounds for IE in terms of n, m, maximum degree, clique number, independence number, and the first Zagreb index. Moreover, we obtain Nordhaus–Gaddum-type results for IE.
Let G be a finite group of order pqr where p > q > r > 2 are prime numbers.
In this paper, we find the spectrum of Cayley graph Cay(G,S) where S ? G \
{e} is a normal symmetric generating subset.
nema
A Laplacian matrix,
L
=
(
ℓ
ij
)
∈
R
n
×
n
, has nonpositive off-diagonal entries and zero row sums. As a matrix associated with a weighted directed graph, it generalizes the Laplacian matrix of an ...ordinary graph. A standardized Laplacian matrix is a Laplacian matrix with
-
1
n
⩽
ℓ
ij
⩽
0
whenever
j
≠
i. We study the spectra of Laplacian matrices and relations between Laplacian matrices and stochastic matrices. We prove that the standardized Laplacian matrices
L
∼
are semiconvergent. The multiplicities of 0 and 1 as the eigenvalues of
L
∼
are equal to the in-forest dimension of the corresponding digraph and one less than the in-forest dimension of the complementary digraph, respectively. We localize the spectra of the standardized Laplacian matrices of order
n and study the asymptotic properties of the corresponding domain. One corollary is that the maximum possible imaginary part of an eigenvalue of
L
∼
converges to
1
π
as
n
→
∞.
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