scholarly journals Merging the A-and Q-spectral theories

2017 ◽  
Vol 11 (1) ◽  
pp. 81-107 ◽  
Author(s):  
V. Nikiforov
Keyword(s):  
Spectral Theory ◽  
Adjacency Matrix ◽  
Diagonal Matrix ◽  
Gradual Change ◽  
Linear Combinations ◽  
Open Problems ◽  
Signless Laplacian

Let G be a graph with adjacency matrix A(G), and let D(G) be the diagonal matrix of the degrees of G: The signless Laplacian Q(G) of G is defined as Q(G):= A(G) +D(G). Cvetkovic called the study of the adjacency matrix the A-spectral theory, and the study of the signless Laplacian{the Q-spectral theory. To track the gradual change of A(G) into Q(G), in this paper it is suggested to study the convex linear combinations A_ (G) of A(G) and D(G) defined by A? (G) := ?D(G) + (1 - ?)A(G), 0 ? ? ? 1. This study sheds new light on A(G) and Q(G), and yields, in particular, a novel spectral Tur?n theorem. A number of open problems are discussed.

scholarly journals Extremal graphs for the sum of the two largest signless Laplacian eigenvalues

2015 ◽  
Vol 30 ◽  
pp. 605-612 ◽  
Author(s):  
Carla Oliveira ◽  
Leonado Lima ◽  
Paula Rama ◽  
Paula Carvalho
Keyword(s):  
Adjacency Matrix ◽  
Simple Graph ◽  
Diagonal Matrix ◽  
Star Graph ◽  
Extremal Graphs ◽  
Laplacian Eigenvalues ◽  
Signless Laplacian ◽  
Additional Edge ◽  
Largest Eigenvalues

Let G be a simple graph on n vertices and e(G) edges. Consider the signless Laplacian, Q(G) = D + A, where A is the adjacency matrix and D is the diagonal matrix of the vertices degree of G. Let q_1(G) and q_2(G) be the first and the second largest eigenvalues of Q(G), respectively, and denote by S_n^+ the star graph with an additional edge. It is proved that inequality q_1(G)+q_2(G) \leq e(G)+3 is tighter for the graph S_n^+ among all firefly graphs and also tighter to S_n^+ than to the graphs K_k \vee K_{n−k} recently presented by Ashraf, Omidi and Tayfeh-Rezaie. Also, it is conjectured that S_n^+ minimizes f(G) = e(G) − q_1(G) − q_2(G) among all graphs G on n vertices.

On the Aα-spectrum of joined union of digraphs

2021 ◽  
pp. 2150086
Author(s):  
Hilal A. Ganie
Keyword(s):  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian ◽  
Adjacency Matrices ◽  
Matrix Formula ◽  
Regular Digraph

Let [Formula: see text] be a digraph of order [Formula: see text] and let [Formula: see text] be the adjacency matrix of [Formula: see text] Let Deg[Formula: see text] be the diagonal matrix of vertex out-degrees of [Formula: see text] For any real [Formula: see text] the generalized adjacency matrix [Formula: see text] of the digraph [Formula: see text] is defined as [Formula: see text] This matrix generalizes the spectral theories of the adjacency matrix and the signless Laplacian matrix of [Formula: see text]. In this paper, we find [Formula: see text]-spectrum of the joined union of digraphs in terms of spectrum of adjacency matrices of its components and the eigenvalues of an auxiliary matrix determined by the joined union. We determine the [Formula: see text]-spectrum of join of two regular digraphs and the join of a regular digraph with the union of two regular digraphs of distinct degrees. As applications, we obtain the [Formula: see text]-spectrum of various families of unsymmetric digraphs.

scholarly journals The normalized distance Laplacian

2021 ◽  
Vol 9 (1) ◽  
pp. 1-18
Author(s):  
Carolyn Reinhart
Keyword(s):  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

scholarly journals Upper bounds on the energy of graphs in terms of matching number

2021 ◽  
pp. 16-16
Author(s):  
Saieed Akbari ◽  
Abdullah Alazemi ◽  
Milica Andjelic
Keyword(s):  
Adjacency Matrix ◽  
Connected Graph ◽  
Short Proof ◽  
Upper Bounds ◽  
Vertex Degree ◽  
Maximum Matching ◽  
Matching Number ◽  
Open Problems

The energy of a graph G, ?(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. The matching number ?(G) is the number of edges in a maximum matching. In this paper, for a connected graph G of order n with largest vertex degree ? ? 6 we present two new upper bounds for the energy of a graph: ?(G) ? (n-1)?? and ?(G) ? 2?(G)??. The latter one improves recently obtained bound ?(G) ? {2?(G)?2?e + 1, if ?e is even; ?(G)(? a + 2?a + ?a-2?a), otherwise, where ?e stands for the largest edge degree and a = 2(?e + 1). We also present a short proof of this result and several open problems.

scholarly journals An Intuitionistic Fuzzy Graph’s Signless Laplacian Energy

2018 ◽  
Vol 7 (4.10) ◽  
pp. 892
Author(s):  
Obbu Ramesh ◽  
S. Sharief Basha
Keyword(s):  
Adjacency Matrix ◽  
Fuzzy Graph ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Graphs ◽  
Laplacian Energy ◽  
Signless Laplacian ◽  
Intuitionistic Fuzzy Graphs

We are extending concept into the Intuitionistic fuzzy graph’ Signless Laplacian energy  instead of the Signless Laplacian energy of fuzzy graph. Now we demarcated an Intuitionistic fuzzy graph’s Signless adjacency matrix and also  an Intuitionistic fuzzy graph’s Signless Laplacian energy. Here we find the Signless Laplacian energy ‘s Intuitionistic fuzzy graphs above and below   boundaries of   an with suitable examples.   

scholarly journals New Bounds for the α-Indices of Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1668
Author(s):  
Eber Lenes ◽  
Exequiel Mallea-Zepeda ◽  
Jonnathan Rodríguez
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Independence Number ◽  
Minimal Degree ◽  
Diagonal Matrix ◽  
The Matrix

Let G be a graph, for any real 0≤α≤1, Nikiforov defines the matrix Aα(G) as Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and diagonal matrix of degrees of the vertices of G. This paper presents some extremal results about the spectral radius ρα(G) of the matrix Aα(G). In particular, we give a lower bound on the spectral radius ρα(G) in terms of order and independence number. In addition, we obtain an upper bound for the spectral radius ρα(G) in terms of order and minimal degree. Furthermore, for n>l>0 and 1≤p≤⌊n−l2⌋, let Gp≅Kl∨(Kp∪Kn−p−l) be the graph obtained from the graphs Kl and Kp∪Kn−p−l and edges connecting each vertex of Kl with every vertex of Kp∪Kn−p−l. We prove that ρα(Gp+1)<ρα(Gp) for 1≤p≤⌊n−l2⌋−1.

scholarly journals Properties of the characteristic polynomials and spectrum of Pn and Cn

2016 ◽  
Vol 5 (2) ◽  
pp. 132
Author(s):  
Essam El Seidy ◽  
Salah Eldin Hussein ◽  
Atef Mohamed
Keyword(s):  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Simple Graph ◽  
Characteristic Polynomials ◽  
Signless Laplacian Matrix ◽  
Path Graph ◽  
Signless Laplacian ◽  
Vertex Set ◽  
Cycle Graph

We consider a finite undirected and connected simple graph  with vertex set  and edge set .We calculated the general formulas of the spectra of a cycle graph and path graph. In this discussion we are interested in the adjacency matrix, Laplacian matrix, signless Laplacian matrix, normalized Laplacian matrix, and seidel adjacency matrix.

scholarly journals Towards a spectral theory of graphs based on the signless Laplacian, I

2009 ◽  
Vol 85 (99) ◽  
pp. 19-33 ◽  
Author(s):  
Dragos Cvetkovic ◽  
Slobodan Simic
Keyword(s):  
Graph Theory ◽  
Spectral Theory ◽  
Spectral Graph Theory ◽  
Line Graphs ◽  
Regular Graphs ◽  
Spectral Graph ◽  
Q Theory ◽  
Signless Laplacian ◽  
M Theory ◽  
Theory Of Graphs

A spectral graph theory is a theory in which graphs are studied by means of eigenvalues of a matrix M which is in a prescribed way defined for any graph. This theory is called M-theory. We outline a spectral theory of graphs based on the signless Laplacians Q and compare it with other spectral theories, in particular with those based on the adjacency matrix A and the Laplacian L. The Q-theory can be composed using various connections to other theories: equivalency with A-theory and L-theory for regular graphs, or with L-theory for bipartite graphs, general analogies with A-theory and analogies with A-theory via line graphs and subdivision graphs. We present results on graph operations, inequalities for eigenvalues and reconstruction problems.

scholarly journals Spectral Sufficient Conditions on Pancyclic Graphs

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Guidong Yu ◽  
Tao Yu ◽  
Xiangwei Xia ◽  
Huan Xu
Keyword(s):  
Spectral Radius ◽  
Spectral Theory ◽  
Sufficient Conditions ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian ◽  
Np Complete ◽  
Pancyclic Graphs ◽  
Theory Of Graphs ◽  
Spectrum Of Graphs

A pancyclic graph of order n is a graph with cycles of all possible lengths from 3 to n . In fact, it is NP-complete that deciding whether a graph is pancyclic. Because the spectrum of graphs is convenient to be calculated, in this study, we try to use the spectral theory of graphs to study this problem and give some sufficient conditions for a graph to be pancyclic in terms of the spectral radius and the signless Laplacian spectral radius of the graph.

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