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Bipartite roots of graphs

2006
*
ACM Transactions on Algorithms
*

In fact, we give a polynomial-time algorithm to count the number

doi:10.1145/1150334.1150337
fatcat:wl6ids57zza3livfea5jzinace
*of*different*bipartite*square*roots**of*a*graph*, although this number could be exponential in the size*of*the input*graph*. ... Finally, we prove the NP-completeness*of*recognizing cubes*of**bipartite**graphs*. ... Cubes*of**Bipartite**Graphs*Since SQUARE*OF**BIPARTITE**GRAPH*is polynomial time solvable, it is natural to ask if we can find a*bipartite*k-th*root**of*a*graph*in polynomial time for k ≥ 3. ...##
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Root polytopes, Tutte polynomials, and a duality theorem for bipartite graphs

2017
*
Proceedings of the London Mathematical Society
*

Let G be a connected

doi:10.1112/plms.12015
fatcat:6uofm2j7xrf3xbdoz7i4irvpfm
*bipartite**graph*with color classes E and V and*root*polytope Q. ... It follows that the interior polynomials*of*(V,E) and its transpose (E,V) agree. When G is a complete*bipartite**graph*, our result recovers a well known hypergeometric identity due to Saalsch\"utz. ... The*root*polytope In this section we recall the general notions*of*f -vector and h-vector and discuss the*root*polytope*of*a*bipartite**graph*. ...##
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The Circuit Polynomial of the Restricted Rooted Product G(Gamma) of Graphs with a Bipartite Core G
[article]

2003
*
arXiv
*
pre-print

As an instance

arXiv:math/0304190v1
fatcat:fikb634uv5brtbkwof3ssu3try
*of*the B-polynomial, the circuit, or cycle, polynomial P(G(Gamma); w)*of*the generalized*rooted*product G(Gamma)*of**graphs*was studied by Farrell and Rosenfeld ( Jour. Math. Sci. ... Herein, we present a new general result and its corollaries concerning the case when the core*graph*G is restricted to be*bipartite*. ... Let T (H) 1 be the restricted*rooted*product*of*a*bipartite**graph*T and an arbitrary*graph*H in which an isomorphic copy*of*H is attached just to every vertex*of*the first (greater) part*of*T . ...##
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Normalized volumes of configurations related with root systems and complete bipartite graphs

2003
*
Discrete Mathematics
*

, C (+) n and D (+) n arising from a complete

doi:10.1016/s0012-365x(02)00690-8
fatcat:3jjbeygfgvfqpcwlumfccknnty
*bipartite**graph*. ... Moreover, the normalized volume*of*the convex hull*of*the subconÿguration*of*A (+) n−1 arising from a complete*bipartite**graph*was computed by Ohsugi and Hibi (Illinois J. ... is a complete*bipartite**graph*on [n]. ...##
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The circuit polynomial of the restricted rooted product G(Γ) of graphs with a bipartite core G

2008
*
Discrete Applied Mathematics
*

*of*the generalized

*rooted*product

*of*

*graphs*, J.

*of*Math. ... As an instance

*of*the B-polynomial, the circuit, or cycle, polynomial P (G( ); w)

*of*the generalized

*rooted*product G( )

*of*

*graphs*was studied by Farrell and Rosenfeld [Block and articulation node polynomials ... Let T (H ) 1 be the restricted

*rooted*product

*of*a

*bipartite*

*graph*T and an arbitrary

*graph*H in which an isomorphic copy

*of*H is attached just to every vertex

*of*the first (greater) part

*of*T. ...

##
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On large bipartite graphs of diameter 3

2013
*
Discrete Mathematics
*

Our approach also bears a proof

doi:10.1016/j.disc.2012.11.013
fatcat:yuu5mc5t3jhg5k3pb5dmk4wlha
*of*the uniqueness*of*the known*bipartite*(5,3,-4)-*graph*, and the non-existence*of**bipartite*(6,3,-4)-*graphs*. ... Here we first present some structural properties*of**bipartite*(d,3,-4)-*graphs*, and later prove there are no*bipartite*(7,3,-4)-*graphs*. ... It is important to mention that the set*of*eigenvalues*of*a*bipartite**graph*is symmetrical with respect to 0. ...##
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Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees
[article]

2014
*
arXiv
*
pre-print

We prove that there exist infinite families

arXiv:1304.4132v2
fatcat:5cs5etacjjf75jzscdu5cdktfi
*of*regular*bipartite*Ramanujan*graphs**of*every degree bigger than 2. ... In particular, we prove the existence*of*infinite families*of*(c,d)-biregular*bipartite**graphs*with all non-trivial eigenvalues bounded by sqrtc-1+sqrtd-1, for all c, d ≥ 3. ... As the 2-lift*of*a*bipartite**graph*is*bipartite*, and the eigenvalues*of*a*bipartite**graph*are symmetric about 0, this 2-lift is also a regular*bipartite*Ramanujan*graph*. ...##
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Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees

2013
*
2013 IEEE 54th Annual Symposium on Foundations of Computer Science
*

We prove that there exist infinite families

doi:10.1109/focs.2013.63
dblp:conf/focs/MarcusSS13
fatcat:3yuulvzxc5a73mxm2a4mkwdgby
*of*regular*bipartite*Ramanujan*graphs**of*every degree bigger than 2. ... In particular, we prove the existence*of*infinite families*of*(c, d)-biregular*bipartite**graphs*with all non-trivial eigenvalues bounded by √ c − 1 + √ d − 1, for all c, d ≥ 3. ... We thank James Lee for suggesting Lemma 6.5 and the simpler proof*of*Theorem 6.6 that appears here. We thank Mirkó Visontai for bringing references [18] , [12] , [9] , and [11] to our attention. ...##
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Graphs with chromatic roots in the interval (1,2)
[article]

2007
*
arXiv
*
pre-print

We present an infinite family

arXiv:0704.2264v1
fatcat:gpduofmhczavbjfi7h3wngv5um
*of*3-connected non-*bipartite**graphs*with chromatic*roots*in the interval (1,2) thus resolving a conjecture*of*Jackson's in the negative. ... In addition, we briefly consider other*graph*classes that are conjectured to have no chromatic*roots*in (1,2). ... A 3-connected*graph*that is not*bipartite**of*odd order has no chromatic*roots*in (1, 2) . ...##
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Interlacing families I: Bipartite Ramanujan graphs of all degrees

2015
*
Annals of Mathematics
*

We prove that there exist infinite families

doi:10.4007/annals.2015.182.1.7
fatcat:xyyn5hqtwzfhtabmgqmbabkm3i
*of*regular*bipartite*Ramanujan*graphs**of*every degree bigger than 2. ... In particular, we prove the existence*of*infinite families*of*(c, d)-biregular*bipartite**graphs*with all non-trivial eigenvalues bounded by √ c − 1 + √ d − 1, for all c, d ≥ 3. ... We thank James Lee for suggesting Lemma 6.5 and the simpler proof*of*Theorem 6.6 that appears here. We thank Mirkó Visontai for bringing references [18] , [12] , [9] , and [11] to our attention. ...##
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Graphs with bounded tree-width and large odd-girth are almost bipartite

2010
*
Journal of combinatorial theory. Series B (Print)
*

We prove that for every k and every ε > 0, there exists g such that every

doi:10.1016/j.jctb.2010.04.004
fatcat:ujqe4fpa75bjjm5bbbi2u4azcy
*graph*with tree-width at most k and odd-girth at least g has circular chromatic number at most 2 + ε. ... By the choice*of*g, the*graph*G 1 has no odd cycle and thus it is a*bipartite**rooted*partial k-tree. ... Lemma 7 . 7 For every k, p and q, there exists a finite number*of*(*bipartite*) types M 1 , . . . , M m such that for any*bipartite**rooted*partial k-tree G with type M, there exists a*bipartite**rooted*partial ...##
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Graphs with bounded tree-width and large odd-girth are almost bipartite
[article]

2009
*
arXiv
*
pre-print

We prove that for every k and every ε>0, there exists g such that every

arXiv:0904.2282v1
fatcat:lhiqcecp5vdvzbvz646yyoyeae
*graph*with tree-width at most k and odd-girth at least g has circular chromatic number at most 2+ε. ... By the choice*of*g, the*graph*G 1 has no odd cycle and thus it is a*bipartite**rooted*partial k-tree. ... To see this, we consider all pairs P = (C, M) where C is a set*of*p-precolorings*of*the*root*and M is a type such that there is a*bipartite**rooted*partial k-tree*of*type M to which no coloring*of*C extends ...##
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On bipartite graphs of defect 2

2009
*
European journal of combinatorics (Print)
*

We find that the eigenvalues other than ±∆

doi:10.1016/j.ejc.2008.09.030
fatcat:hw24rjuutza6vepnhvrkpeileu
*of*such*graphs*are the*roots**of*the polynomials H D−1 (x) ± 1, where H D−1 (x) is the Dickson polynomial*of*the second kind with parameter ∆ − 1 and degree D ... It is known that the Moore*bipartite*bound provides an upper bound on the order*of*a connected*bipartite**graph*. ... For D = 2, the Moore*bipartite**graphs*are the complete*bipartite**graphs**of*degree ∆, while for D = 3, 4 and 6, they are the incidence*graphs**of*projective planes*of*order ∆ − 1,*of*generalized quadrangles ...##
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Feedback vertex set on chordal bipartite graphs
[article]

2012
*
arXiv
*
pre-print

Let G=(A,B,E) be a

arXiv:1104.3915v2
fatcat:kdk5iod57bam7d5mm4j5kacxlm
*bipartite**graph*with color classes A and B. The*graph*G is chordal*bipartite*if G has no induced cycle*of*length more than four. Let G=(V,E) be a*graph*. ... We show that the feedback vertex set problem can be solved in polynomial time on chordal*bipartite**graphs*. ... Separators in chordal*bipartite**graphs*In this section we analyze the structure*of*chordal*bipartite*by means*of*minimal separators. Definition 5. Let G = (V, E) be a*graph*. ...##
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Graphs with Chromatic Roots in the Interval $(1,2)$

2007
*
Electronic Journal of Combinatorics
*

We present an infinite family

doi:10.37236/1019
fatcat:wbx3pjgmibgvvelbcky6fxsiom
*of*3-connected non-*bipartite**graphs*with chromatic*roots*in the interval $(1,2)$ thus resolving a conjecture*of*Jackson's in the negative. ... In addition, we briefly consider other*graph*classes that are conjectured to have no chromatic*roots*in $(1,2)$. ... A 3-connected*graph*that is not*bipartite**of*odd order has no chromatic*roots*in (1, 2) . ...
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