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.

The spectral characterizations of the connected multicone graphs Kw ▽ LHS and Kw ▽ LGQ(3,9)

2018 ◽  
Vol 10 (02) ◽  
pp. 1850019 ◽  
Author(s):  
Ali Zeydi Abdian ◽  
S. Morteza Mirafzal
Keyword(s):  
Optimization Problems ◽  
Spectral Graph Theory ◽  
Combinatorial Optimization Problems ◽  
Spectral Graph ◽  
The Past ◽  
Laplacian Spectra ◽  
Signless Laplacian ◽  
Multicone Graph ◽  
Spectral Characterizations ◽  
Sims Graph

In the past decades, graphs that are determined by their spectrum have received much more and more attention, since they have been applied to several fields, such as randomized algorithms, combinatorial optimization problems and machine learning. An important part of spectral graph theory is devoted to determining whether given graphs or classes of graphs are determined by their spectra or not. So, finding and introducing any class of graphs which are determined by their spectra can be an interesting and important problem. The main aim of this study is to characterize two classes of multicone graphs which are determined by their adjacency, Laplacian and signless Laplacian spectra. A multicone graph is defined to be the join of a clique and a regular graph. Let [Formula: see text] denote a complete graph on [Formula: see text] vertices. In the paper, we show that multicone graphs [Formula: see text] and [Formula: see text] are determined by both their adjacency spectra and their Laplacian spectra, where [Formula: see text] and [Formula: see text] denote the Local Higman–Sims graph and the Local [Formula: see text] graph, respectively. In addition, we prove that these multicone graphs are determined by their signless Laplacian spectra.

scholarly journals Nordhaus–Gaddum-Type Relations for Arithmetic-Geometric Spectral Radius and Energy

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Yajing Wang ◽  
Yubin Gao
Keyword(s):  
Graph Theory ◽  
Spectral Radius ◽  
Adjacency Matrix ◽  
Upper Bounds ◽  
Simple Graph ◽  
Spectral Graph Theory ◽  
Extremal Graphs ◽  
Spectral Graph ◽  
Vertex Set

Spectral graph theory plays an important role in engineering. Let G be a simple graph of order n with vertex set V=v1,v2,…,vn. For vi∈V, the degree of the vertex vi, denoted by di, is the number of the vertices adjacent to vi. The arithmetic-geometric adjacency matrix AagG of G is defined as the n×n matrix whose i,j entry is equal to di+dj/2didj if the vertices vi and vj are adjacent and 0 otherwise. The arithmetic-geometric spectral radius and arithmetic-geometric energy of G are the spectral radius and energy of its arithmetic-geometric adjacency matrix, respectively. In this paper, some new upper bounds on arithmetic-geometric energy are obtained. In addition, we present the Nordhaus–Gaddum-type relations for arithmetic-geometric spectral radius and arithmetic-geometric energy and characterize corresponding extremal graphs.

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