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### Bipartite density of cubic graphs

Abraham Berman, Xiao-Dong Zhang
2003 Discrete Mathematics
We ÿrst obtain the exact value for bipartite density of a cubic line graph on n vertices.  ...  Then we give an upper bound for the bipartite density of cubic graphs in terms of the smallest eigenvalue of the adjacency matrix.  ...   to our attention, which results in a new short proof of Theorem 2.4.  ...

### Bipartite density of triangle-free subcubic graphs

Xuding Zhu
2009 Discrete Applied Mathematics
A graph is subcubic if its maximum degree is at most 3. The bipartite density of a graph G is defined as b(G) = max{|E(B)|/|E(G)| : B is a bipartite subgraph of G}.  ...  It was conjectured by Bondy and Locke that if G is a triangle-free subcubic graph, then b(G) ≥ 4 5 and equality holds only if G is in a list of seven small graphs.  ...  They further proved that the Petersen graph and the dodecahedron (see Fig. 1 ) are the only triangle-free cubic graphs with bipartite density 4 5 .  ...

### Disjoint Triangles of a Cubic Line Graph

Xiao-Dong Zhang, Stanislaw Bylka
2004 Graphs and Combinatorics
Moreover, the equitable chromatic number, acyclic chromatic number and bipartite density of G are 3, 3, 7 9 respectively.  ...  In this paper, we prove that a cubic line graph G on n vertices rather than the complete graph K 4 has b n 3 c vertex-disjoint triangles and the vertex independence number b n 3 c.  ...  The authors are grateful to the referees for valuable suggestions and comments which results in a great improvement of the original manuscript.  ...

### On a bipartition problem of Bollobás and Scott

Baogang Xu, Xingxing Yu
2009 Combinatorica
The bipartite density of a graph G is max{|E(H)|/|E(G)| : H is a bipartite subgraph of G}. It is NP-hard to determine the bipartite density of any triangle-free cubic graph.  ...  Let H denote the collection of all connected cubic graphs which have bipartite density 4 5 and contain biased maximum bipartite subgraphs. Bollobás and Scott asked which cubic graphs belong to H.  ...  bipartite density 4 5 .  ...

### Dense Graph Partitioning on sparse and dense graphs [article]

Cristina Bazgan, Katrin Casel, Pierre Cazals
2022 arXiv   pre-print
In particular, we prove that it is NP-hard on dense bipartite graphs as well as on cubic graphs.  ...  We consider the problem of partitioning a graph into a non-fixed number of non-overlapping subgraphs of maximum density.  ...  The density d(G n,m ) of a complete bipartite graph G n,m is greater than or equal to the density d(P) of any partition P of G n,m . Proof.  ...

### Page 8942 of Mathematical Reviews Vol. , Issue 2003m [page]

2003 Mathematical Reviews
Summary: “We first obtain the exact value for bipartite density of a cubic line graph on n vertices.  ...  | (IL-TECH; Haifa) Bipartite density of cubic graphs. (English summary) Discrete Math. 260 (2003), no. 1-3, 27-35.  ...

### LDPC codes constructed from cubic symmetric graphs [article]

Dean Crnkovic, Sanja Rukavina, Marina Simac
2020 arXiv   pre-print
Information on the LDPC codes constructed from bipartite cubic symmetric graphs with less than 200 vertices is presented as well.  ...  In this paper we present a construction of LDPC codes from cubic symmetric graphs.  ...  In this paper we construct LDPC codes using bipartite cubic symmetric graphs as Tanner graphs of LDPC codes.  ...

### Page 6549 of Mathematical Reviews Vol. , Issue 93m [page]

1993 Mathematical Reviews
Given vertex-disjoint graphs G,; and G> of order n, and given a k-regular bipartite graph B having the two sets of vertices for its bipartition, we call G; U G2UB a generalized prism A;(G, G2) over G;  ...  Summary: “The density of a nontrivial graph G is defined by g(G) =|E(G)|/(V(G) — @(G)), where w@(G) denotes the number of components of G.  ...

### Page 5669 of Mathematical Reviews Vol. , Issue 88k [page]

1988 Mathematical Reviews
Several easy variations of the general problem are considered. In particular, the paper supplies an O(n + m) algorithm to find a minimum m density matching of size n of any complete bipartite graph.  ...  On bipartite matchings of minimum density. J. Algorithms 8 (1987), no. 4, 480-502. Suppose two parallel horizontal lines are given, r and s, say. Also, suppose P [resp.  ...

### Page 1724 of Mathematical Reviews Vol. , Issue 87d [page]

1987 Mathematical Reviews
Suppose z denotes the independence ratio of a graph G, the pro- portion of vertices in a largest independent set in G, and b denotes the bipartite density, the proportion of edges in a largest bipartite  ...  The result is potentially useful for deducing properties of the chromatic polynomial of a graph. E. J. Farrell (Trinidad) Locke, Stephen C. (1-FLAT) Bipartite density and the independence ratio. J.  ...

### Eigenvalue spacings for regular graphs [article]

D. Jakobson, S.D.Miller, I.Rivin, Z. Rudnick
2003 arXiv   pre-print
We carry out a numerical study of fluctuations in the spectrum of regular graphs.  ...  A review of the basic facts on graphs and their spectra is included.  ...  (a) a cubic graph on 2000 vertices, (b) a 5-valent graph on 500 vertices. Figure 2 . 2 Level spacing distribution of a cubic graph on 2000 vertices against GOE.  ...

### Triangle-free subcubic graphs with minimum bipartite density

Baogang Xu, Xingxing Yu
2008 Journal of combinatorial theory. Series B (Print)
It is an NP-hard problem to determine the bipartite density of any given trianglefree cubic graph.  ...  dodecahedron are the only triangle-free cubic graphs with bipartite density 4 5 .  ...  dodecahedron (shown in Fig. 1 ) are the only cubic graphs with bipartite density 4 5  ...

### Eigenvalue Spacings for Regular Graphs [chapter]

Dmitry Jakobson, Stephen D. Miller, Igor Rivin, Zeév Rudnick
1999 IMA Volumes in Mathematics and its Applications
We carry out a numerical study of fluctuations in the spectrum of regular graphs.  ...  A review of the basic facts on graphs and their spectra is included.  ...  Figure 4 . 4 Distribution of normalized discrepancies for cubic graphs vs.  ...

### Page 9585 of Mathematical Reviews Vol. , Issue 2004m [page]

2004 Mathematical Reviews
Mohan (New Delhi) 05C Graph theory 2004m:05172 2004m:05169 05C50 Stevanovic, Dragan (YU-NISSM; Nis) Bipartite density of cubic graphs: the case of equality.  ...  Zhang [Discrete Math. 260 (2003), no. 1-3, 27-35; MR 2003m:05096] obtained an upper bound for the bipartite density of cubic graphs in terms of the smallest eigenvalue of an adjacency matrix and attempted  ...

### Page 3520 of Mathematical Reviews Vol. , Issue 84i [page]

1984 Mathematical Reviews
Ellingham [“Non-Hamiltonian 3-connected cubic bipartite graphs”, Research Report No. 28, Dept. of Math., Univ.  ...  Bipartite cubic graphs and a shortness exponent. Discrete Math. 44 (1983), no. 3, 327-330. The author constructs a non-Hamiltonian bipartite 3-connected graph with only 78 vertices.  ...
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