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1
Content available remote

On Spectra Of Variants Of The Corona Of Two Graphs And Some New Equienergetic Graphs

100%
Adiga C., Rakshith B. R.
Discussiones Mathematicae Graph Theory
|
2016
|
tom 36
|
nr 1
127-140
EN
Let G and H be two graphs. The join G ∨ H is the graph obtained by joining every vertex of G with every vertex of H. The corona G ○ H is the graph obtained by taking one copy of G and |V (G)| copies of H and joining the i-th vertex of G to every vertex in the i-th copy of H. The neighborhood corona G★H is the graph obtained by taking one copy of G and |V (G)| copies of H and joining the neighbors of the i-th vertex of G to every vertex in the i-th copy of H. The edge corona G ◇ H is the graph obtained by taking one copy of G and |E(G)| copies of H and joining each terminal vertex of i-th edge of G to every vertex in the i-th copy of H. Let G1, G2, G3 and G4 be regular graphs with disjoint vertex sets. In this paper we compute the spectrum of (G1 ∨ G2) ∪ (G1 ★ G3), (G1 ∨ G2) ∪ (G2 ★ G3) ∪ (G1 ★ G4), (G1 ∨G2)∪(G1 ○G3), (G1 ∨G2)∪(G2 ○G3)∪(G1 ○G4), (G1 ∨G2)∪(G1 ◇G3), (G1 ∨ G2) ∪ (G2 ◇ G3) ∪ (G1 ◇ G4), (G1 ∨ G2) ∪ (G2 ○ G3) ∪ (G1 ★ G3), (G1 ∨ G2) ∪ (G2 ○ G3) ∪ (G1 ◇ G4) and (G1 ∨ G2) ∪ (G2 ★ G3) ∪ (G1 ◇ G4). As an application, we show that there exist some new pairs of equienergetic graphs on n vertices for all n ≥ 11.
2
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Construction of Cospectral Integral Regular Graphs

100%
Bapat R. B., Karimi M.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 37
|
nr 3
595-609
EN
Graphs G and H are called cospectral if they have the same characteristic polynomial. If eigenvalues are integral, then corresponding graphs are called integral graph. In this article we introduce a construction to produce pairs of cospectral integral regular graphs. Generalizing the construction of G4(a, b) and G5(a, b) due to Wang and Sun, we define graphs 𝒢4(G,H) and 𝒢5(G,H) and show that they are cospectral integral regular when G is an integral q-regular graph of order m and H is an integral q-regular graph of order (b − 2)m for some integer b ≥ 3.
3
Content available remote

A note on distance spectral radius of trees

100%
Wang Y., Xing R., Zhou B., Dong F.
Special Matrices
|
2017
|
tom 5
|
nr 1
296-300
EN
The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. We determine the unique non-starlike non-caterpillar tree with maximal distance spectral radius.
4
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Tricyclic graphs with exactly two main eigenvalues

100%
Fan X., Luo Y., Gao X.
Open Mathematics
|
2013
|
tom 11
|
nr 10
1800-1816
EN
An eigenvalue of a graph G is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. Let G 0 be the graph obtained from G by deleting all pendant vertices and δ(G) the minimum degree of vertices of G. In this paper, all connected tricyclic graphs G with δ(G 0) ≥ 2 and exactly two main eigenvalues are determined.
5
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On the Laplacian Coefficients of Tricyclic Graphs with Prescribed Matching Number

81%
Luo J., Zhu Z., Wan R.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 37
|
nr 3
505-522
EN
Let [...] φ(L(G))=det (xI−L(G))=∑k=0n(−1)kck(G)xn−k $\phi (L(G)) = \det (xI - L(G)) = \sum\nolimits_{k = 0}^n {( - 1)^k c_k (G)x^{n - k} } $ be the Laplacian characteristic polynomial of G. In this paper, we characterize the minimal graphs with the minimum Laplacian coefficients in 𝒢n,n+2(i) (the set of all tricyclic graphs with fixed order n and matching number i). Furthermore, the graphs with the minimal Laplacian-like energy, which is the sum of square roots of all roots on ϕ(L(G)), is also determined in 𝒢n,n+2(i).
6
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On the Spectral Characterizations of Graphs

81%
Huang J., Li S.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 37
|
nr 3
729-744
EN
Several matrices can be associated to a graph, such as the adjacency matrix or the Laplacian matrix. The spectrum of these matrices gives some informations about the structure of the graph and the question “Which graphs are determined by their spectrum?” is still a difficult problem in spectral graph theory. Let [...] 𝒰p2q ${\cal U}_p^{2q}$ be the set of graphs obtained from Cp by attaching two pendant edges to each of q (q ⩽ p) vertices on Cp, whereas [...] 𝒱p2q ${\cal V}_p^{2q}$ the subset of [...] 𝒰p2q ${\cal U}_p^{2q}$ with odd p and its q vertices of degree 4 being nonadjacent to each other. In this paper, we show that each graph in [...] 𝒰p2q ${\cal U}_p^{2q}$ , p even and its q vertices of degree 4 being consecutive, is determined by its Laplacian spectrum. As well we show that if G is a graph without isolated vertices and adjacency cospectral with the graph in [...] 𝒱pp−1={H} ${\cal V}_p^{p - 1} = \{ H\}$ , then G ≅ H.
7
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A formula for all minors of the adjacency matrix and an application

62%
Bapat R. B., Lal A. K., Pati S.
Special Matrices
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2014
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tom 2
|
nr 1
EN
We supply a combinatorial description of any minor of the adjacency matrix of a graph. This description is then used to give a formula for the determinant and inverse of the adjacency matrix, A(G), of a graph G, whenever A(G) is invertible, where G is formed by replacing the edges of a tree by path bundles.
8
Content available remote

Gromov hyperbolic cubic graphs

62%
Pestana D., Rodríguez J., Sigarreta J., Villeta M.
Open Mathematics
|
2012
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tom 10
|
nr 3
1141-1151
EN
If X is a geodesic metric space and x 1; x 2; x 3 ∈ X, a geodesic triangle T = {x 1; x 2; x 3} is the union of the three geodesics [x 1 x 2], [x 2 x 3] and [x 3 x 1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. We denote by δ(X) the sharp hyperbolicity constant of X, i.e., δ(X) = inf {δ ≥ 0: X is δ-hyperbolic}. We obtain information about the hyperbolicity constant of cubic graphs (graphs with all of their vertices of degree 3), and prove that for any graph G with bounded degree there exists a cubic graph G* such that G is hyperbolic if and only if G* is hyperbolic. Moreover, we prove that for any cubic graph G with n vertices, we have δ(G) ≤ min {3n/16 + 1; n/4}. We characterize the cubic graphs G with δ(G) ≤ 1. Besides, we prove some inequalities involving the hyperbolicity constant and other parameters for cubic graphs.
9
Content available remote

The maximum multiplicity and the two largest multiplicities of eigenvalues in a Hermitian matrix whose graph is a tree

62%
Fernandes R.
Special Matrices
|
2015
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tom 3
|
nr 1
EN
The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree, M1, was understood fully (froma combinatorial perspective) by C.R. Johnson, A. Leal-Duarte (Linear Algebra and Multilinear Algebra 46 (1999) 139-144). Among the possible multiplicity lists for the eigenvalues of Hermitian matrices whose graph is a tree, we focus upon M2, the maximum value of the sum of the two largest multiplicities when the largest multiplicity is M1. Upper and lower bounds are given for M2. Using a combinatorial algorithm, cases of equality are computed for M2.
10
Content available remote

Enumeration of spanning trees in the sequence of Dürer graphs

62%
Li S.
Open Mathematics
|
2017
|
tom 15
|
nr 1
1591-1598
EN
In this paper, we calculate the number of spanning trees in the sequence of Dürer graphs with a special feature that it has two alternate states. Using the electrically equivalent transformations, we obtain the weights of corresponding equivalent graphs and further derive relationships for spanning trees between the Dürer graphs and transformed graphs. By algebraic calculations, we obtain a closed-form formula for the number of spanning trees with regard to iteration step. Finally we compare the entropy of our graph with other studied graphs and see that its value of entropy lies in the interval of those of graphs with average degree being 3 and 4.
11
Content available remote

Structures ofW(2.2) Lie conformal algebra

52%
Yuan L., Wu H.
Open Mathematics
|
2016
|
tom 14
|
nr 1
629-640
EN
The purpose of this paper is to study W(2, 2) Lie conformal algebra, which has a free ℂ[∂]-basis {L, M} such that [...] [LλL]=(∂+2λ)L,[LλM]=(∂+2λ)M,[MλM]=0 $\begin{equation}[{L_\lambda }L] = (\partial + 2\lambda )L,[{L_\lambda }M] = (\partial + 2\lambda )M,[{M_\lambda }M] = 0]\end{equation}$ . In this paper, we study conformal derivations, central extensions and conformal modules for this Lie conformal algebra. Also, we compute the cohomology of this Lie conformal algebra with coefficients in its modules. In particular, we determine its cohomology with trivial coefficients both for the basic and reduced complexes.
12
Content available remote

On the number of spanning trees, the Laplacian eigenvalues, and the Laplacian Estrada index of subdivided-line graphs

52%
Shang Y.
Open Mathematics
|
2016
|
tom 14
|
nr 1
641-648
EN
As a generalization of the Sierpiński-like graphs, the subdivided-line graph Г(G) of a simple connected graph G is defined to be the line graph of the barycentric subdivision of G. In this paper we obtain a closed-form formula for the enumeration of spanning trees in Г(G), employing the theory of electrical networks. We present bounds for the largest and second smallest Laplacian eigenvalues of Г(G) in terms of the maximum degree, the number of edges, and the first Zagreb index of G. In addition, we establish upper and lower bounds for the Laplacian Estrada index of Г(G) based on the vertex degrees of G. These bounds are also connected with the number of spanning trees in Г(G).
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