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Minimizing algebraic connectivity over connected graphs with fixed girth

2002
*
Discrete Mathematics
*

Let Gn;g denote the class of all

doi:10.1016/s0012-365x(01)00355-7
fatcat:drklknyfgrg2zmc6ujwe6dt6ri
*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 ...##
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On the homogeneous algebraic graphs of large girth and their applications

2009
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Linear Algebra and its Applications
*

We consider some results on such

doi:10.1016/j.laa.2008.08.023
fatcat:dlt53j7ywzcq5enffof4age6lu
*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] . ...##
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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*. ...##
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Extremizing algebraic connectivity subject to graph theoretic constraints

1998
*
The Electronic Journal of Linear Algebra
*

Analogous results are proved for unicyclic

doi:10.13001/1081-3810.1014
fatcat:vmfq6sfd3rd4fcnayt46lnxymy
*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 ...##
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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. ...##
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On the implementation of cryptoalgorithms based on algebraic graphs over some commutative rings

2008
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Condensed Matter Physics
*

The paper is devoted to computer implementation of some

doi:10.5488/cmp.11.2.347
fatcat:afdr65zbyzalxbz4vrmwutleim
*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. ...##
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Ordering trees and graphs with few cycles by algebraic connectivity

2014
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Linear Algebra and its Applications
*

Hence, a large number of papers about ordering

doi:10.1016/j.laa.2014.06.016
fatcat:zlkujdy7jfct3c36vxhxutgyeq
*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 ...##
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A Heawood-type result for the algebraic connectivity of graphs on surfaces
[article]

2001
*
arXiv
*
pre-print

More precisely, it is shown that the

arXiv:math/0109191v1
fatcat:sldd75uwcjagtn3mdjj7kgkbku
*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. ...##
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On the Comparison of Cryptographical Properties of Two Different Families of Graphs with Large Cycle Indicator

2012
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Mathematics in Computer Science
*

The paper is devoted to the implementations of the public key algorithms based on simple

doi:10.1007/s11786-012-0121-x
fatcat:ichszy5crjdrtjzt5bv43xtuzu
*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 ...##
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Old and new results on algebraic connectivity of graphs

2007
*
Linear Algebra and its Applications
*

Hence, this could not be different

doi:10.1016/j.laa.2006.08.017
fatcat:aorfqxv6enewreyflkpeumvzgy
*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 . ...##
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Minimizing Laplacian spectral radius of unicyclic graphs with fixed girth
[article]

2010
*
arXiv
*
pre-print

In this paper we consider the following problem:

arXiv:1012.0888v2
fatcat:h4wkyljdtjgmdlbgytcmub3kva
*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 ...##
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Some bounds on the modified Randic index

2015
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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,

doi:10.5937/kgjsci1537079m
fatcat:qnqtndgkvrh2jo5ckghcba47wy
*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). ...##
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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. ...##
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Ramanujan Graphs with Small Girth
[article]

2003
*
arXiv
*
pre-print

We construct an infinite family of (q+1)-regular Ramanujan

arXiv:math/0306196v1
fatcat:tbk576xxefhzbbqvvof3cc6nxe
*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 ∞). ...##
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Ramanujan Graphs with Small Girth

2003
*
Combinatorica
*

We construct an infinite family of (q +1)−regular Ramanujan

doi:10.1007/s00493-003-0029-9
fatcat:hsceplsr75fpzkvussemgwkbkm
*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 ∞). ...
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