scholarly journals Unicyclic reflexive graphs with seven loaded vertices of the cycle

Filomat ◽  
2009 ◽  
Vol 23 (3) ◽  
pp. 257-268
Author(s):  
Tamara Koledin ◽  
Zoran Radosavljevic
Keyword(s):  
Adjacency Matrix ◽  
Simple Graph ◽  
Largest Eigenvalue ◽  
Reflexive Graph ◽  
Second Largest Eigenvalue

A simple graph is reflexive if the second largest eigenvalue of its (0, 1)- adjacency matrix does not exceed 2. A vertex of the cycle of unicyclic simple graph is said to be loaded if its degree is greater than 2. In this paper we establish that the length of the cycle of unicyclic reflexive graph with seven loaded vertices is at most 10 and find all such graphs with the length of the cycle 10, 9 and 8.

scholarly journals Some graph mappings that preserve the sign of λ2 - r

2017 ◽  
Vol 11 (1) ◽  
pp. 148-165
Author(s):  
Bojana Mihailovic ◽  
Marija Rasajski
Keyword(s):  
Adjacency Matrix ◽  
Simple Graph ◽  
Equivalence Relations ◽  
Cyclic Structure ◽  
Largest Eigenvalue ◽  
Second Largest Eigenvalue

In this article we deal with the sign of ?2 - r, r > 0, where ?2 is the second largest eigenvalue of (adjacency matrix of) a simple graph and present some methods of determining it for some classes of graphs. The main result is a set of graph mappings that preserve the value of sgn (?2 - r). These mappings induce equivalence relations among involved graphs, thus providing a way to indirectly apply the GRS-theorem (the generalization of so-called RS-theorem) to some GRS-undecidable (or RS-undecidable) graphs. To present possible applications, we revisit some of the previous results for reexive graphs (graphs whose second largest eigenvalue does not exceed 2). We show how maximal reexive graphs that belong to various families depending on their cyclic structure, can be reduced to RS-decidable graphs in terms of corresponding equivalence relations.

scholarly journals NONEXISTENCE OF A CIRCULANT EXPANDER FAMILY

2010 ◽  
Vol 83 (1) ◽  
pp. 87-95
Author(s):  
KA HIN LEUNG ◽  
VINH NGUYEN ◽  
WASIN SO
Keyword(s):  
Regular Graph ◽  
Elementary Proof ◽  
Explicit Construction ◽  
Simple Graph ◽  
Regular Graphs ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Largest Eigenvalue ◽  
Image Position ◽  
Second Largest Eigenvalue

AbstractThe expansion constant of a simple graph G of order n is defined as where $E(S, \overline {S})$ denotes the set of edges in G between the vertex subset S and its complement $\overline {S}$. An expander family is a sequence {Gi} of d-regular graphs of increasing order such that h(Gi)>ϵ for some fixed ϵ>0. Existence of such families is known in the literature, but explicit construction is nontrivial. A folklore theorem states that there is no expander family of circulant graphs only. In this note, we provide an elementary proof of this fact by first estimating the second largest eigenvalue of a circulant graph, and then employing Cheeger’s inequalities where G is a d-regular graph and λ2(G) denotes the second largest eigenvalue of G. Moreover, the associated equality cases are discussed.

Eigenpolytopes of Distance Regular Graphs

1998 ◽  
Vol 50 (4) ◽  
pp. 739-755 ◽  
Author(s):  
C. D. Godsil
Keyword(s):  
Convex Hull ◽  
Orthogonal Projection ◽  
Adjacency Matrix ◽  
Gram Matrix ◽  
Regular Graphs ◽  
Largest Eigenvalue ◽  
The Matrix ◽  
Vertex Set ◽  
Distance Regular Graphs ◽  
Second Largest Eigenvalue

AbstractLet X be a graph with vertex set V and let A be its adjacency matrix. If E is the matrix representing orthogonal projection onto an eigenspace of A with dimension m, then E is positive semi-definite. Hence it is the Gram matrix of a set of |V| vectors in Rm. We call the convex hull of a such a set of vectors an eigenpolytope of X. The connection between the properties of this polytope and the graph is strongest when X is distance regular and, in this case, it is most natural to consider the eigenpolytope associated to the second largest eigenvalue of A. The main result of this paper is the characterisation of those distance regular graphs X for which the 1-skeleton of this eigenpolytope is isomorphic to X.

Some spectral properties of Aα-matrix

2019 ◽  
Vol 11 (06) ◽  
pp. 1950070
Author(s):  
Shuang Zhang ◽  
Yan Zhu
Keyword(s):  
Real Number ◽  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Spectral Properties ◽  
Upper Bounds ◽  
Largest Eigenvalue ◽  
Number Formula ◽  
Second Largest Eigenvalue ◽  
Spectral Radius Formula

For a real number [Formula: see text], the [Formula: see text]-matrix of a graph [Formula: see text] is defined to be [Formula: see text] where [Formula: see text] and [Formula: see text] are the adjacency matrix and degree diagonal matrix of [Formula: see text], respectively. The [Formula: see text]-spectral radius of [Formula: see text], denoted by [Formula: see text], is the largest eigenvalue of [Formula: see text]. In this paper, we consider the upper bound of the [Formula: see text]-spectral radius [Formula: see text], also we give some upper bounds for the second largest eigenvalue of [Formula: see text]-matrix.

scholarly journals On the Spectrum of Laplacian Matrix

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Akbar Jahanbani ◽  
Seyed Mahmoud Sheikholeslami ◽  
Rana Khoeilar
Keyword(s):  
Lower Bounds ◽  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Simple Graph ◽  
Upper And Lower Bounds ◽  
Largest Eigenvalue ◽  
Smallest Eigenvalue ◽  
The Matrix ◽  
Vertex Degrees ◽  
The Largest Eigenvalue

Let G be a simple graph of order n . The matrix ℒ G = D G − A G is called the Laplacian matrix of G , where D G and A G denote the diagonal matrix of vertex degrees and the adjacency matrix of G , respectively. Let l 1 G , l n − 1 G be the largest eigenvalue, the second smallest eigenvalue of ℒ G respectively, and λ 1 G be the largest eigenvalue of A G . In this paper, we will present sharp upper and lower bounds for l 1 G and l n − 1 G . Moreover, we investigate the relation between l 1 G and λ 1 G .

scholarly journals A class of reflexive cactuses with four cycles

2003 ◽  
pp. 64-85 ◽  
Author(s):  
Zoran Radosavljevic ◽  
Marija Rasajski
Keyword(s):  
Simple Graph ◽  
Common Vertex ◽  
Complete Case ◽  
Largest Eigenvalue ◽  
Treelike Graph ◽  
Second Largest Eigenvalue

A simple graph is reflexive if its second largest eigenvalue ?2 is less than or equal to 2. A graph is a cactus, or a treelike graph, if any pair of its cycles (circuits) has at most one common vertex. For a lot of cactuses the property ?2 ? 2 can be tested by identifying and deleting a single cut-vetex (Theorem 1). if this theorem cannot be applied to a connected reflexive cactus and if all its cycles do not form a bundle, such a graph has at most five cycles. On the same conditions, in this paper we find some classes of maximal reflexive cactuses with four cycles. The complete case of four cycles, together with that of five cycles, is being settled in [10].

scholarly journals A note on a walk-based inequality for the index of a signed graph

2021 ◽  
Vol 9 (1) ◽  
pp. 19-21
Author(s):  
Zoran Stanić
Keyword(s):  
Adjacency Matrix ◽  
Signed Graph ◽  
Arbitrary Length ◽  
Largest Eigenvalue ◽  
The Largest Eigenvalue

Abstract We derive an inequality that includes the largest eigenvalue of the adjacency matrix and walks of an arbitrary length of a signed graph. We also consider certain particular cases.

Graphs with the second signless Laplacian eigenvalue ≤ 4

2021 ◽  
Vol 10 (1) ◽  
pp. 131-152
Author(s):  
Stephen Drury
Keyword(s):  
General Solution ◽  
Knapsack Problem ◽  
Laplacian Eigenvalue ◽  
Simple Graphs ◽  
Largest Eigenvalue ◽  
Signless Laplacian ◽  
Second Largest Eigenvalue

Abstract We discuss the question of classifying the connected simple graphs H for which the second largest eigenvalue of the signless Laplacian Q(H) is ≤ 4. We discover that the question is inextricable linked to a knapsack problem with infinitely many allowed weights. We take the first few steps towards the general solution. We prove that this class of graphs is minor closed.

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.

On the estimation of the second largest eigenvalue of Markov ciphers

2016 ◽  
pp. n/a-n/a
Author(s):  
Weijia Xue ◽  
Tingting Lin ◽  
Xin Shun ◽  
Fenglei Xue ◽  
Xuejia Lai
Keyword(s):  
Largest Eigenvalue ◽  
Second Largest Eigenvalue
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