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### Bipartite roots of graphs

Lap Chi Lau
2006 ACM Transactions on Algorithms
In fact, we give a polynomial-time algorithm to count the number 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.  ...

### Root polytopes, Tutte polynomials, and a duality theorem for bipartite graphs

Tamás Kálmán, Alexander Postnikov
2017 Proceedings of the London Mathematical Society
Let G be a connected 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.  ...

### 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 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 .  ...

### Normalized volumes of configurations related with root systems and complete bipartite graphs

Hidefumi Ohsugi, Takayuki Hibi
2003 Discrete Mathematics
, C (+) n and D (+) n arising from a complete 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].  ...

### 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.  ...

### On large bipartite graphs of diameter 3

Ramiro Feria-Purón, Mirka Miller, Guillermo Pineda-Villavicencio
2013 Discrete Mathematics
Our approach also bears a proof 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.  ...

### Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees [article]

Adam Marcus, Daniel A. Spielman, Nikhil Srivastava
2014 arXiv   pre-print
We prove that there exist infinite families 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.  ...

### Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees

Adam Marcus, Daniel A. Spielman, Nikhil Srivastava
2013 2013 IEEE 54th Annual Symposium on Foundations of Computer Science
We prove that there exist infinite families 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.  ...

### Graphs with chromatic roots in the interval (1,2) [article]

Gordon F. Royle
2007 arXiv   pre-print
We present an infinite family 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) .  ...

### Interlacing families I: Bipartite Ramanujan graphs of all degrees

Adam Marcus, Daniel Spielman, Nikhil Srivastava
2015 Annals of Mathematics
We prove that there exist infinite families 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.  ...

### Graphs with bounded tree-width and large odd-girth are almost bipartite

Alexandr V. Kostochka, Daniel Král', Jean-Sébastien Sereni, Michael Stiebitz
2010 Journal of combinatorial theory. Series B (Print)
We prove that for every k and every ε > 0, there exists g such that every 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  ...

### Graphs with bounded tree-width and large odd-girth are almost bipartite [article]

Alexandr V. Kostochka, Daniel Kral', Jean-Sebastien Sereni, Michael Stiebitz
2009 arXiv   pre-print
We prove that for every k and every ε>0, there exists g such that every 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  ...

### On bipartite graphs of defect 2

Charles Delorme, Leif K. Jørgensen, Mirka Miller, Guillermo Pineda-Villavicencio
2009 European journal of combinatorics (Print)
We find that the eigenvalues other than ±∆ 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  ...

### Feedback vertex set on chordal bipartite graphs [article]

Ton Kloks, Ching-Hao Liu, Sheung-Hung Poon
2012 arXiv   pre-print
Let G=(A,B,E) be a 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.  ...

### Graphs with Chromatic Roots in the Interval \$(1,2)\$

Gordon F. Royle
2007 Electronic Journal of Combinatorics
We present an infinite family 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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