scholarly journals Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

2021 ◽  
Author(s):  
Jürgen Jost ◽  
Raffaella Mulas ◽  
Florentin Münch
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Spectral Gap ◽  
Complete Bipartite Graph ◽  
Graph Laplacian ◽  
New Method ◽  
Single Edge ◽  
Largest Eigenvalue ◽  
Complement Graph ◽  
The Largest Eigenvalue

AbstractWe offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least $$\frac{n+1}{n-1}$$ n + 1 n - 1 provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $$\frac{n-1}{2}$$ n - 1 2 . With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $$\frac{n-1}{2}$$ n - 1 2 .

scholarly journals Spectral Extrema for Graphs: The Zarankiewicz Problem

10.37236/212 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
László Babai ◽  
Barry Guiduli
Keyword(s):  
Bipartite Graph ◽  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Complete Bipartite Graph ◽  
Average Degree ◽  
Largest Eigenvalue ◽  
Zarankiewicz Problem ◽  
The Largest Eigenvalue

Let $G$ be a graph on $n$ vertices with spectral radius $\lambda$ (this is the largest eigenvalue of the adjacency matrix of $G$). We show that if $G$ does not contain the complete bipartite graph $K_{t ,s}$ as a subgraph, where $2\le t \le s$, then $$\lambda \le \Big((s-1)^{1/t }+o(1)\Big)n^{1-1/t }$$ for fixed $t$ and $s$ while $n\to\infty$. Asymptotically, this bound matches the Kővári-Turán-Sós upper bound on the average degree of $G$ (the Zarankiewicz problem).

On the Crossing Number of $K_{m,n}$

10.37236/1748 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Nagi H. Nahas
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number

The best lower bound known on the crossing number of the complete bipartite graph is : $$cr(K_{m,n}) \geq (1/5)(m)(m-1)\lfloor n/2 \rfloor \lfloor(n-1)/2\rfloor$$ In this paper we prove that: $$cr(K_{m,n}) \geq (1/5)m(m-1)\lfloor n/2 \rfloor \lfloor (n-1)/2 \rfloor + 9.9 \times 10^{-6} m^2n^2$$ for sufficiently large $m$ and $n$.

scholarly journals Lower bound for the spectral radius of the starlike trees

2021 ◽  
Vol 2090 (1) ◽  
pp. 012127
Author(s):  
Rubí Arrizaga-Zercovich
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Adjacency Matrix ◽  
Maximum Degree ◽  
Largest Eigenvalue ◽  
Acyclic Graph ◽  
Starlike Tree ◽  
The Largest Eigenvalue

Abstract A tree is a connected acyclic graph. A tree is called a starlike if exactly one of its vertices has degree greater than two. Let λι be the largest eigenvalue of the adjacency matrix of a starlike tree. In this work, we obtain a lower bound for the spectral radius of a starlike tree. This bound only depends of the maximum degree of the vertices.

TENACITY OF TOTAL GRAPHS

2014 ◽  
Vol 25 (05) ◽  
pp. 553-562 ◽  
Author(s):  
YINKUI LI ◽  
ZONGTIAN WEI ◽  
XIAOKUI YUE ◽  
ERQIANG LIU
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Communication Networks ◽  
Interconnection Networks ◽  
Complete Bipartite Graph ◽  
Total Graph ◽  
The Stability

Communication networks must be constructed to be as stable as possible, not only with the respect to the initial disruption, but also with respect to the possible reconstruction. Many graph theoretical parameters have been used to describe the stability of communication networks. Tenacity is a reasonable one, which shows not only the difficulty to break down the network but also the damage that has been caused. Total graphs are the largest graphs formed by the adjacent relations of elements of a graph. Thus, total graphs are highly recommended for the design of interconnection networks. In this paper, we determine the tenacity of the total graph of a path, cycle and complete bipartite graph, and thus give a lower bound of the tenacity for the total graph of a graph.

ENHANCING THE SYNCHRONIZABILITY OF SCALE-FREE NETWORKS BY ADDING EDGES

2010 ◽  
Vol 21 (01) ◽  
pp. 67-77 ◽  
Author(s):  
SHENG-JUN WANG ◽  
ZHI-XI WU ◽  
HAI-RONG DONG ◽  
GUANRONG CHEN
Keyword(s):  
Nearest Neighbors ◽  
Laplacian Matrix ◽  
New Method ◽  
Scale Free Network ◽  
Largest Eigenvalue ◽  
Scale Free ◽  
Scale Free Networks ◽  
Free Network ◽  
The Largest Eigenvalue

To efficiently enhance the synchronizability of a scale-free network by adding some edges, we numerically study the effect of edge-adding on the spectrum of the network Laplacian matrix. Based on the relation between the largest eigenvalue of the Laplacian matrix and the largest degree of the scale-free network, we show that adding a new edge to the node of largest degree will generally weaken the synchronizability of a scale-free network. We consequently propose a method to effectively enhance the network synchronizability by attaching the new edge to a node whose nearest-neighbors have small degrees. The effect of the new method is analyzed and demonstrated with comparisons.

scholarly journals The rank of Pseudo walk matrices: controllable and recalcitrant pairs

2020 ◽  
Vol 3 (3) ◽  
pp. 41-52
Author(s):  
Alexander Farrugia ◽  
Keyword(s):  
Lower Bound ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Minimal Polynomial ◽  
Full Rank ◽  
Gram Matrix ◽  
Largest Eigenvalue ◽  
The Largest Eigenvalue

A pseudo walk matrix \(\mathbf{W}_\mathbf{v}\) of a graph \(G\) having adjacency matrix \(\mathbf{A}\) is an \(n\times n\) matrix with columns \(\mathbf{v},\mathbf{A}\mathbf{v},\mathbf{A}^2\mathbf{v},\ldots,\mathbf{A}^{n-1}\mathbf{v}\) whose Gram matrix has constant skew diagonals, each containing walk enumerations in \(G\). We consider the factorization over \(\mathbb{Q}\) of the minimal polynomial \(m(G,x)\) of \(\mathbf{A}\). We prove that the rank of \(\mathbf{W}_\mathbf{v}\), for any walk vector \(\mathbf{v}\), is equal to the sum of the degrees of some, or all, of the polynomial factors of \(m(G,x)\). For some adjacency matrix \(\mathbf{A}\) and a walk vector \(\mathbf{v}\), the pair \((\mathbf{A},\mathbf{v})\) is controllable if \(\mathbf{W}_\mathbf{v}\) has full rank. We show that for graphs having an irreducible characteristic polynomial over \(\mathbb{Q}\), the pair \((\mathbf{A},\mathbf{v})\) is controllable for any walk vector \(\mathbf{v}\). We obtain the number of such graphs on up to ten vertices, revealing that they appear to be commonplace. It is also shown that, for all walk vectors \(\mathbf{v}\), the degree of the minimal polynomial of the largest eigenvalue of \(\mathbf{A}\) is a lower bound for the rank of \(\mathbf{W}_\mathbf{v}\). If the rank of \(\mathbf{W}_\mathbf{v}\) attains this lower bound, then \((\mathbf{A},\mathbf{v})\) is called a recalcitrant pair. We reveal results on recalcitrant pairs and present a graph having the property that \((\mathbf{A},\mathbf{v})\) is neither controllable nor recalcitrant for any walk vector \(\mathbf{v}\).

scholarly journals Edit Distance and its Computation

10.37236/744 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
József Balogh ◽  
Ryan Martin
Keyword(s):  
Bipartite Graph ◽  
Edit Distance ◽  
Complete Bipartite Graph ◽  
New Method ◽  
Independent Interest ◽  
Regularity Lemma ◽  
Hereditary Property ◽  
Vertex Set ◽  
Hereditary Properties ◽  
Szemeredi's Regularity Lemma

In this paper, we provide a method for determining the asymptotic value of the maximum edit distance from a given hereditary property. This method permits the edit distance to be computed without using Szemerédi's Regularity Lemma directly. Using this new method, we are able to compute the edit distance from hereditary properties for which it was previously unknown. For some graphs $H$, the edit distance from ${\rm Forb}(H)$ is computed, where ${\rm Forb}(H)$ is the class of graphs which contain no induced copy of graph $H$. Those graphs for which we determine the edit distance asymptotically are $H=K_a+E_b$, an $a$-clique with $b$ isolated vertices, and $H=K_{3,3}$, a complete bipartite graph. We also provide a graph, the first such construction, for which the edit distance cannot be determined just by considering partitions of the vertex set into cliques and cocliques. In the process, we develop weighted generalizations of Turán's theorem, which may be of independent interest.

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