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Minimizing algebraic connectivity over connected graphs with fixed girth

Shaun M. Fallat, Steve Kirkland, Sukanta Pati
2002 Discrete Mathematics  
Let Gn;g denote the class of all connected graphs on n vertices with ÿxed girth g.  ...  We prove that if n ¿ 3g − 1, then the graph which uniquely minimizes the algebraic connectivity over Gn;g is the unicyclic "lollipop" graph Cn;g obtained by appending a g cycle to a pendant vertex of a  ...  Among all connected graphs on n vertices with ÿxed girth g, algebraic connectivity is minimized by a unicyclic graph of girth g, with the following property: there are at most two connected components  ... 
doi:10.1016/s0012-365x(01)00355-7 fatcat:drklknyfgrg2zmc6ujwe6dt6ri

On the homogeneous algebraic graphs of large girth and their applications

T. Shaska, V. Ustimenko
2009 Linear Algebra and its Applications  
We consider some results on such algebraic graphs over any field. The upper bound on the dimension of variety of edges for algebraic graphs of girth 2d is established.  ...  We consider the problem of constructing homogeneous algebraic graphs with a prescribed girth and formulate some problems motivated by classical extremal graph theory.  ...  The first family of connected algebraic graphs over F q of a large girth and arbitrarily large degree had been constructed in [23] .  ... 
doi:10.1016/j.laa.2008.08.023 fatcat:dlt53j7ywzcq5enffof4age6lu

Page 4630 of Mathematical Reviews Vol. , Issue 2000g [page]

2000 Mathematical Reviews  
In particular, the unique minimizer and maximizer of the algebraic connectivity are given over all such graphs with girth 3.” 2000g:05102 05C50 05C75 Lee, Yueh-Shin (RC-NCT-AM; Hsinchu) ; Chang, G.  ...  Analogous results are proved for unicyclic graphs with fixed girth.  ... 

Extremizing algebraic connectivity subject to graph theoretic constraints

Shaun Fallat, Steve Kirkland
1998 The Electronic Journal of Linear Algebra  
Analogous results are proved for unicyclic graphs with xed girth. In particular, the unique minimizer and maximizer of the algebraic connectivity is given over all such graphs with girth 3.  ...  When the radius of a graph is the speci ed constraint the unique minimizer of the algebraic connectivity o ver all such graphs is also determined.  ...  Among all connected g r aphs on n vertices with xed girth s, the algebraic connectivity is minimized by a unicyclic graph with girth s, with the following property: There a r e at most two connected c  ... 
doi:10.13001/1081-3810.1014 fatcat:vmfq6sfd3rd4fcnayt46lnxymy

Page 1602 of Mathematical Reviews Vol. , Issue 2003C [page]

2003 Mathematical Reviews  
, SK); Kirkland, Steve (3-RGN; Regina, SK); Pati, Sukanta (3-RGN; Regina, SK) Minimizing algebraic connectivity over connected graphs with fixed girth.  ...  Two of the authors conjectured earlier that C,¢ is the unique graph minimizing the algebraic connectivity over G,,¢ and confirmed this conjecture for g = 3 [see S. M. Fallat and S. J.  ... 

On the implementation of cryptoalgorithms based on algebraic graphs over some commutative rings

Kotorowicz, Ustimenko
2008 Condensed Matter Physics  
The paper is devoted to computer implementation of some graph based stream ciphers. We compare the time performance of this new algorithm with fast, but no very secure RC4, and with DES.  ...  The software package with new encryption algorithms is ready for the demonstration.  ...  If the minimal size of the connected component of each G i is growing with i, then the encryption scheme is not a block cipher but a stream cipher.  ... 
doi:10.5488/cmp.11.2.347 fatcat:afdr65zbyzalxbz4vrmwutleim

Ordering trees and graphs with few cycles by algebraic connectivity

Nair Abreu, Claudia Marcela Justel, Oscar Rojo, Vilmar Trevisan
2014 Linear Algebra and its Applications  
Hence, a large number of papers about ordering graphs by algebraic connectivity, mainly about trees and graphs with few cycles, have been published.  ...  By considering the fact that the algebraic connectivity is related to the connectivity and shape of the graphs, several structural properties of graphs relative to this parameter have been studied.  ...  The authors are grateful to Carlos Hoppen for helping us to precisely define a probabilistic version of the problem of relating the algebraic connectivity and diameter and to João Carvalho, who set up  ... 
doi:10.1016/j.laa.2014.06.016 fatcat:zlkujdy7jfct3c36vxhxutgyeq

A Heawood-type result for the algebraic connectivity of graphs on surfaces [article]

Pedro Freitas
2001 arXiv   pre-print
More precisely, it is shown that the algebraic connectivity of a surface S, defined as the supremum of a(G) over all graphs that can be embedded in S, is equal to the chromatic number of S.  ...  We prove that the algebraic connectivity a(G) of a graph embedded on a nonplanar surface satisfies a Heawood-type result.  ...  We shall now restrict our attention to certain classes of graphs. An immediate consequence of Theorem 5.1 is a bound for the asymptotic connectivity of regular graphs with a given fixed girth g.  ... 
arXiv:math/0109191v1 fatcat:sldd75uwcjagtn3mdjj7kgkbku

On the Comparison of Cryptographical Properties of Two Different Families of Graphs with Large Cycle Indicator

Michał Klisowski, Vasyl Ustimenko
2012 Mathematics in Computer Science  
The paper is devoted to the implementations of the public key algorithms based on simple algebraic graphs A(n, K ) and D(n, K ) defined over the same finite commutative ring K .  ...  with nonzero coefficients) in the comparison with algorithms based on D(n, F q ), q = 2 i , i = 8, 16, 32, 64.  ...  Conclusion Results of computer simulation show that multivariate cryptosystem corresponding to family of graphs A(n, F q ), q = 2 i , i = 8, 16, 32, 64 have a better density (number of monomial expressions  ... 
doi:10.1007/s11786-012-0121-x fatcat:ichszy5crjdrtjzt5bv43xtuzu

Old and new results on algebraic connectivity of graphs

Nair Maria Maia de Abreu
2007 Linear Algebra and its Applications  
Hence, this could not be different with respect to the algebraic connectivity, as described in [4, 22, 47, 49] .  ...  For example, it is well-known that a graph is connected if and only if its algebraic connectivity is different from zero.  ...  Among all connected graphs on n vertices with fixed girth s, the algebraic connectivity is minimized by the lollipop graph C n,s .  ... 
doi:10.1016/j.laa.2006.08.017 fatcat:aorfqxv6enewreyflkpeumvzgy

Minimizing Laplacian spectral radius of unicyclic graphs with fixed girth [article]

Kamal Lochan Patra, Binod Kumar Sahoo
2010 arXiv   pre-print
In this paper we consider the following problem: Over the class of all simple connected unicyclic graphs on n vertices with girth g (n,g being fixed), which graph minimizes the Laplacian spectral radius  ...  We prove that the graph U_n,g (defined in Section 1) uniquely minimizes the Laplacian spectral radius for n≥ 2g-1 when g is even and for n≥ 3g-1 when g is odd.  ...  As far as the class of connected unicyclic graphs on n vertices with fixed girth is concerned, the problem of minimizing and maximizing the algebraic connectivity has been studied in [1] and [2] , respectively  ... 
arXiv:1012.0888v2 fatcat:h4wkyljdtjgmdlbgytcmub3kva

Some bounds on the modified Randic index

Hemmasi Mahsa, Iranmanesh Ali
2015 Kragujevac Journal of Science  
In this paper, we present some new lower and upper bounds for the modified Randic index in terms of maximum, minimum degree, girth, algebraic connectivity, diameter and average distance.  ...  Also we obtained relations between this index with Harmonic and Atom-bond connectivity indices. Finally, as an application we computed this index for some classes of nano-structures and linear chains.  ...  Partial support by the Center of Excellence of Algebraic Hyper structures and its Applications of Tarbiat Modares University (CEAHA) is gratefully acknowledged by the second author (AI).  ... 
doi:10.5937/kgjsci1537079m fatcat:qnqtndgkvrh2jo5ckghcba47wy

Page 492 of Mathematical Reviews Vol. , Issue 82b [page]

1982 Mathematical Reviews  
An algebraic treatment of the case e=2, g=2r>8 yields the following nice result. Theorem: There is no regular graph G with girth 2r>8 and excess 2.  ...  The k-connectedness of unlabelled J. London Math. Soc. (2) 18 (1978), no. 3, 397-402. Let U(k,q) denote the number of unlabelled k-connected graphs with q lines and no isolated points.  ... 

Ramanujan Graphs with Small Girth [article]

Yair Glasner
2003 arXiv   pre-print
We construct an infinite family of (q+1)-regular Ramanujan graphs X_n of girth 1.  ...  We also give covering maps X_n+1 --> X_n such that the minimal common covering of all the graphs is the universal covering tree.  ...  Ramanujan Graphs of Small Girth Let H = H u,v be a definite quaternion algebra defined over Q (0 < u, v ∈ Q, i 2 = −u, j 2 = −v, ij = −ji = k, H ramifies at ∞).  ... 
arXiv:math/0306196v1 fatcat:tbk576xxefhzbbqvvof3cc6nxe

Ramanujan Graphs with Small Girth

Yair Glasner
2003 Combinatorica  
We construct an infinite family of (q +1)−regular Ramanujan graphs X n of girth 1.  ...  We also give covering maps X n+1 → X n such that the minimal common covering of all the X n 's is the universal covering tree.  ...  Ramanujan Graphs of Small Girth Let H = H u,v be a definite quaternion algebra defined over Q (0 < u, v ∈ Q, i 2 = −u, j 2 = −v, ij = −ji = k, H ramifies at ∞).  ... 
doi:10.1007/s00493-003-0029-9 fatcat:hsceplsr75fpzkvussemgwkbkm
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