New families of integral graphs

2016 ◽  
Vol 08 (04) ◽  
pp. 1650063
Author(s):  
Indulal Gopalapillai
Keyword(s):  
Adjacency Matrix ◽  
Simple Graph ◽  
Matrix Formula ◽  
Integral Graphs

Let [Formula: see text] be a simple graph with an adjacency matrix [Formula: see text]. Then the eigenvalues of [Formula: see text] are the eigenvalues of [Formula: see text] and form the spectrum, [Formula: see text] of [Formula: see text]. The graph [Formula: see text] is integral if [Formula: see text] consists of only integers. In this paper, we define three new operations on graphs and characterize all integral graphs in the resulting families. The resulting families are denoted by [Formula: see text], and [Formula: see text]. These characterizations allow us to exhibit many new infinite families of integral graphs.

A note on the relationship between graph energy and determinant of adjacency matrix

2019 ◽  
Vol 11 (01) ◽  
pp. 1950001
Author(s):  
Igor Ž. Milovanović ◽  
Emina I. Milovanović ◽  
Marjan M. Matejić ◽  
Akbar Ali
Keyword(s):  
Lower Bounds ◽  
Adjacency Matrix ◽  
Bipartite Graphs ◽  
Simple Graph ◽  
Graph Invariants ◽  
Graph Energy ◽  
The Matrix ◽  
Matrix Formula ◽  
The Relationship

Let [Formula: see text] be a simple graph of order [Formula: see text], without isolated vertices. Denote by [Formula: see text] the adjacency matrix of [Formula: see text]. Eigenvalues of the matrix [Formula: see text], [Formula: see text], form the spectrum of the graph [Formula: see text]. An important spectrum-based invariant is the graph energy, defined as [Formula: see text]. The determinant of the matrix [Formula: see text] can be calculated as [Formula: see text]. Recently, Altindag and Bozkurt [Lower bounds for the energy of (bipartite) graphs, MATCH Commun. Math. Comput. Chem. 77 (2017) 9–14] improved some well-known bounds on the graph energy. In this paper, several inequalities involving the graph invariants [Formula: see text] and [Formula: see text] are derived. Consequently, all the bounds established in the aforementioned paper are improved.

scholarly journals SPEKTRUM PADA GRAF REGULER KUAT

2013 ◽  
Vol 5 (1) ◽  
pp. 13
Author(s):  
Rizki Mulyani ◽  
Triyani Triyani ◽  
Niken Larasati
Keyword(s):  
Regular Graph ◽  
Adjacency Matrix ◽  
Strongly Regular Graph ◽  
Simple Graph ◽  
Eigen Value ◽  
Strongly Regular

This article studied spectrum of strongly regular graph. This spectrum can be determined by the number of walk with lenght l on connected simple graph, equation of square adjacency matrix and eigen value of strongly regular graph.

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 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.

scholarly journals On the Main Eigenvalues of Universal Adjacency Matrices and U-Controllable Graphs

2015 ◽  
Vol 30 ◽  
pp. 812-826
Author(s):  
Alexander Farrugia ◽  
Irene Sciriha
Keyword(s):  
Adjacency Matrix ◽  
Sufficient Conditions ◽  
Laplacian Matrix ◽  
Simple Graph ◽  
Identity Matrix ◽  
The Matrix ◽  
Vertex Degrees ◽  
Definition Of ◽  
Main Eigenvalues ◽  
Necessary And Sufficient

A universal adjacency matrix U of a graph G is a linear combination of the 0–1 adjacency matrix A, the diagonal matrix of vertex degrees D, the identity matrix I and the matrix J each of whose entries is 1. A main eigenvalue of U is an eigenvalue having an eigenvector that is not orthogonal to the all–ones vector. It is shown that the number of distinct main eigenvalues of U associated with a simple graph G is at most the number of orbits of any automorphism of G. The definition of a U–controllable graph is given using control–theoretic techniques and several necessary and sufficient conditions for a graph to be U–controllable are determined. It is then demonstrated that U–controllable graphs are asymmetric and that the converse is false, showing that there exist both regular and non–regular asymmetric graphs that are not U–controllable for any universal adjacency matrix U. To aid in the discovery of these counterexamples, a gamma–Laplacian matrix L(gamma) is used, which is a simplified form of U. It is proved that any U-controllable graph is a L(gamma)–controllable graph for some parameter gamma.

scholarly journals Laplacian integral graphs with a given degree sequence constraint

2021 ◽  
Vol 40 (6) ◽  
pp. 1431-1448
Author(s):  
Ansderson Fernandes Novanta ◽  
Carla Silva Oliveira ◽  
Leonardo de Lima
Keyword(s):  
Adjacency Matrix ◽  
Degree Sequence ◽  
Laplacian Matrix ◽  
Bipartite Graphs ◽  
Diagonal Matrix ◽  
Connected Graphs ◽  
Sequence Constraint ◽  
The Matrix ◽  
Vertex Degrees ◽  
Integral Graphs

Let G be a graph on n vertices. The Laplacian matrix of G, denoted by L(G), is defined as L(G) = D(G) −A(G), where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the vertex degrees of G. A graph G is said to be L-integral if all eigenvalues of the matrix L(G) are integers. In this paper, we characterize all Lintegral non-bipartite graphs among all connected graphs with at most two vertices of degree larger than or equal to three.

scholarly journals Some classes of integral graphs which belong to the class αKa U βKb,b

2003 ◽  
Vol 74 (88) ◽  
pp. 25-36 ◽  
Author(s):  
Mirko Lepovic
Keyword(s):  
Simple Graph ◽  
Largest Eigenvalue ◽  
Integral Graphs ◽  
The Largest Eigenvalue

Let G be a simple graph and let G denote its complement. We say that G is integral if its spectrum consists of integral values. We have recently established a characterization of integral graphs which belong to the class ?Ka U ?Kb,b, where mG denotes the m-fold union of the graph G. In this work we investigate integral graphs from the class ?Ka U ?Kb,b with ?1 = a+b where ?1 is the largest eigenvalue of ?Ka U ?Kb,b.

scholarly journals Some Algebraic Properties of a Class of Integral Graphs Determined by Their Spectrum

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Jia-Bao Liu ◽  
S. Morteza Mirafzal ◽  
Ali Zafari
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Adjacency Matrix ◽  
Prime Integer ◽  
Integral Graph ◽  
Algebraic Properties ◽  
Integral Graphs

Let Γ = V , E be a graph. If all the eigenvalues of the adjacency matrix of the graph Γ are integers, then we say that Γ is an integral graph. A graph Γ is determined by its spectrum if every graph cospectral to it is in fact isomorphic to it. In this paper, we investigate some algebraic properties of the Cayley graph Γ = Cay ℤ n , S , where n = p m ( p is a prime integer and m ∈ ℕ ) and S = a ∈ ℤ n | a , n = 1 . First, we show that Γ is an integral graph. Also, we determine the automorphism group of Γ . Moreover, we show that Γ and K v ▽ Γ are determined by their spectrum.

scholarly journals Constructing Fifteen Infinite Classes of Nonregular Bipartite Integral Graphs

10.37236/732 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Ligong Wang ◽  
Cornelis Hoede
Keyword(s):  
Number Theory ◽  
Adjacency Matrix ◽  
Diophantine Equations ◽  
Matrix Theory ◽  
Discrete Math ◽  
Computer Search ◽  
Characteristic Polynomials ◽  
Integral Property ◽  
Open Problems ◽  
Integral Graphs

A graph is called integral if all its eigenvalues (of the adjacency matrix) are integers. In this paper, the graphs $S_1(t)=K_{1,t}$, $S_2(n,t)$, $S_3(m,n,t)$, $S_4(m,n,p,q)$, $S_5(m,n)$, $S_6(m,n,t)$, $S_8(m,n)$, $S_9(m,n,p,q)$, $S_{10}(n)$, $S_{13}(m,n)$, $S_{17}(m, n, p, q)$, $S_{18}(n,p,q,t)$, $S_{19}(m,n,p,t)$, $S_{20}(n,p,q)$ and $S_{21}(m,t)$ are defined. We construct the fifteen classes of larger graphs from the known 15 smaller integral graphs $S_1-S_6$, $S_8-S_{10}$, $S_{13}$, $S_{17}-S_{21}$ (see also Figures 4 and 5, Balińska and Simić, Discrete Math. 236(2001) 13-24). These classes consist of nonregular and bipartite graphs. Their spectra and characteristic polynomials are obtained from matrix theory. We obtain their integral property by using number theory and computer search. All these classes are infinite. They are different from those in the literature. It is proved that the problem of finding such integral graphs is equivalent to solving Diophantine equations. We believe that it is useful for constructing other integral graphs. The discovery of these integral graphs is a new contribution to the search of integral graphs. Finally, we propose several open problems for further study.

scholarly journals A graph theoretical approach to cis/trans isomerism

2014 ◽  
Vol 79 (7) ◽  
pp. 805-813 ◽  
Author(s):  
Boris Furtula ◽  
Giorgi Lekishvili ◽  
Ivan Gutman
Keyword(s):  
Graph Theory ◽  
Electron Energy ◽  
Theoretical Approach ◽  
Adjacency Matrix ◽  
Molecular Graph ◽  
Energy Difference ◽  
Simple Graph ◽  
Total Electron ◽  
Total Electron Energy ◽  
Graph Theoretical Approach

A simple graph-theory-based model is put forward, by means of which it is possible to express the energy difference between geometrically non-equivalent forms of a conjugated polyene. This is achieved by modifying the adjacency matrix of the molecular graph, and including into it information on cis/trans constellations. The total ?-electron energy thus calculated is in excellent agreement with the enthalpies of the underlying isomers and conformers.

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