scholarly journals On Consensus of Star-Composed Networks with an Application of Laplacian Spectrum

2017 ◽  
Vol 2017 ◽  
pp. 1-13
Author(s):  
Da Huang ◽  
Haijun Jiang ◽  
Zhiyong Yu ◽  
Qiongxiang Huang ◽  
Xing Chen
Keyword(s):  
Communication Network ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Laplacian Matrix ◽  
Convergence Speed ◽  
Laplacian Spectrum ◽  
Graph Spectra ◽  
Nonzero Eigenvalue ◽  
Insight Into

In this paper, we mainly study the performance of star-composed networks which can achieve consensus. Specifically, we investigate the convergence speed and robustness of the consensus of the networks, which can be measured by the smallest nonzero eigenvalue λ2 of the Laplacian matrix and the H2 norm of the graph, respectively. In particular, we introduce the notion of the corona of two graphs to construct star-composed networks and apply the Laplacian spectrum to discuss the convergence speed and robustness for the communication network. Finally, the performances of the star-composed networks have been compared, and we find that the network in which the centers construct a balanced complete bipartite graph has the most advantages of performance. Our research would provide a new insight into the combination between the field of consensus study and the theory of graph spectra.

scholarly journals The Smith and Critical Groups of the Square Rook's Graph and its Complement

10.37236/5442 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Joshua E. Ducey ◽  
Jonathan Gerhard ◽  
Noah Watson
Keyword(s):  
Normal Form ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Line Graph ◽  
Cartesian Product ◽  
Laplacian Matrix ◽  
Critical Group ◽  
Critical Groups ◽  
Complete Graphs ◽  
Vertex Set

Let $R_{n}$ denote the graph with vertex set consisting of the squares of an $n \times n$ grid, with two squares of the grid adjacent when they lie in the same row or column.  This is the square rook's graph, and can also be thought of as the Cartesian product of two complete graphs of order $n$, or the line graph of the complete bipartite graph $K_{n,n}$.  In this paper we compute the Smith group and critical group of the graph $R_{n}$ and its complement.  This is equivalent to determining the Smith normal form of both the adjacency and Laplacian matrix of each of these graphs.  In doing so we verify a 1986 conjecture of Rushanan.

Graceful Labeling for Some Complete Bipartite Graph

2018 ◽  
Vol 9 (12) ◽  
pp. 2147-2152
Author(s):  
V. Raju ◽  
M. Paruvatha vathana
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Graceful Labeling

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 Join Products K2,3 + Cn

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 925
Author(s):  
Michal Staš
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number ◽  
Arithmetic Means ◽  
Minimum Number ◽  
Edge Crossings

The crossing number cr ( G ) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main goal of the paper is to state the crossing number of the join product K 2 , 3 + C n for the complete bipartite graph K 2 , 3 , where C n is the cycle on n vertices. In the proofs, the idea of a minimum number of crossings between two distinct configurations in the various forms of arithmetic means will be extended. Finally, adding one more edge to the graph K 2 , 3 , we also offer the crossing number of the join product of one other graph with the cycle C n .

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 An Exact Turán Result for Tripartite 3-Graphs

10.37236/5203 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Adam Sanitt ◽  
John Talbot
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Maximum Size ◽  
Unique Graph

Mantel's theorem says that among all triangle-free graphs of a given order the balanced complete bipartite graph is the unique graph of maximum size. We prove an analogue of this result for 3-graphs. Let $K_4^-=\{123,124,134\}$, $F_6=\{123,124,345,156\}$ and $\mathcal{F}=\{K_4^-,F_6\}$: for $n\neq 5$ the unique $\mathcal{F}$-free 3-graph of order $n$ and maximum size is the balanced complete tripartite 3-graph $S_3(n)$ (for $n=5$ it is $C_5^{(3)}=\{123,234,345,145,125\}$). This extends an old result of Bollobás that $S_3(n) $ is the unique 3-graph of maximum size with no copy of $K_4^-=\{123,124,134\}$ or $F_5=\{123,124,345\}$.

scholarly journals Two-Distance-Primitive Graphs

10.37236/8890 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Wei Jin ◽  
Ci Xuan Wu ◽  
Jin Xin Zhou
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Complete Classification ◽  
Second Step ◽  
Transitive Graph ◽  
Vertex Transitive ◽  
Vertex Transitive Graph ◽  
Cyclic Graph ◽  
Acts 2

A 2-distance-primitive graph is a vertex-transitive graph whose vertex stabilizer is primitive on both the first step and the second step neighborhoods. Let $\Gamma$ be such a graph. This paper shows that either $\Gamma$ is a cyclic graph, or $\Gamma$ is a complete bipartite graph, or $\Gamma$ has girth at most $4$ and the vertex stabilizer acts faithfully on both the first step and the second step neighborhoods. Also a complete classification is given of such graphs  satisfying that the vertex stabilizer acts $2$-transitively on the second step neighborhood. Finally, we determine the unique 2-distance-primitive graph which is  locally cyclic.

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