## Growth in Products of Graphs (2001)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Pisanski01growthin,

author = {Tomaz Pisanski and Thomas W. Tucker},

title = {Growth in Products of Graphs},

year = {2001}

}

### OpenURL

### Abstract

We present some results on the growth in various products of graphs.

### Citations

1123 |
Graph Theory
- Harary
- 1995
(Show Context)
Citation Context ...G; x) orjust∆(x) andΓ(x). Note that δ(Gv,n)=γ(Gv,n) − γ(Gv,n− 1),n>0. Recall that for a vertex u in a finite graph G the eccentricity of u is the maximum distance from u to any other vertex of G; see =-=[10]-=-. Let d denote the eccentricity of u and let n be the number of vertices of G. Since the series for Γ(v; x) has only a finite number of non-constant terms, we can introduce a polynomial h(v; x) thatco... |

39 |
de la Harpe, On problems related to growth, entropy, and spectrum
- Grigorchuk, P
(Show Context)
Citation Context ... case of vertex-transitive factors. 1 Introduction Some basic facts on growth of graphs are presented. Although the concept is most useful for vertex-transitive graphs and comes primarily from groups =-=[9]-=-, we consider finite or special classes of locally finite graphs. The emphasis is on the actual computation of generating functions for various operations on graphs, where we try to express the growth... |

24 | A conjecture concerning a limit of non-Cayley graphs
- Diestel, Leader
(Show Context)
Citation Context ...dded sequence of rooted graphs P0 ⊂ Q1 ⊂ P1 ⊂ Q2 ⊂ ... whose limit graph is denoted by Gr ∗Hp. It is not hard to see that Hp ∗Gr = Gr ∗Hp. The concept of the limit graph can be found, for instance in =-=[1, 5]-=-. Example 4.1 Consider K3 ∗ K3. Label the vertices in the first copy of K3 by r, 0, 1 and the vertices in the second copy by r ′ , 0 ′ and 1 ′ .Letr and r ′ be the corresponding roots. The vertices,or... |

23 | Growth functions on Fuchsian groups and the Euler characteristic - Floyd, Plotnick - 1987 |

8 | Computer algebra libraries for combinatorial structures
- Flajolet, Salvy
- 1995
(Show Context)
Citation Context ...e try to express the growth series of various products in terms of growth series of their factors. Many of these operations can be carried our completely automatically, as shown by Flajolet and Salvy =-=[6]-=-. All graphs in this paper are simplicial and connected, unless specified otherwise. ∗ Supported in part by “Ministrstvo za znanost in tehnologijo Republike Slovenije”, program no. 101-511 and proj. n... |

7 | Growth in repeated truncations of maps, Atti Sem - Pisanski, Tucker |

6 |
Vertex-transitive graphs and vertex-transitive maps
- Babai
- 1991
(Show Context)
Citation Context ...dded sequence of rooted graphs P0 ⊂ Q1 ⊂ P1 ⊂ Q2 ⊂ ... whose limit graph is denoted by Gr ∗Hp. It is not hard to see that Hp ∗Gr = Gr ∗Hp. The concept of the limit graph can be found, for instance in =-=[1, 5]-=-. Example 4.1 Consider K3 ∗ K3. Label the vertices in the first copy of K3 by r, 0, 1 and the vertices in the second copy by r ′ , 0 ′ and 1 ′ .Letr and r ′ be the corresponding roots. The vertices,or... |

6 | An exact sequence for rings of polynomials in partly commuting indeterminates - Dicks - 1981 |

3 | Growth functions for semi-regular tilings of the hyperbolic plane, Geom. Dedicata 53 - Floyd, Plotnick - 1994 |

3 |
Rcomp: A mathematica package for computing with recursive sequences
- Nemes, Petkovˇsek
- 1995
(Show Context)
Citation Context ...positive integer d the distance between any two vertices of G ⊠ H is at most d if and only if the distance in both projections is at most d. � There is a program by Marko Petkovˇsek based on the work =-=[17]-=- included in the system Vega [20] for calculating the Hadamard product of any two rational functions. Corollary 5.2 Let G and H be growth regular then their strong product G ⊠ H is growth regular. 6 L... |

3 |
Counterexamples involving growth series and the Euler characteristic
- Parry
- 1988
(Show Context)
Citation Context ...; x) − 1)F2(x) F2(x) =1+(∆(Hp; x) − 1)F1(x) By elimination of F1(x) andF2(x) one obtains the formula stated in the theorem. � The above theorem was used for groups in several places, see for instance =-=[19]-=-. In the special case when G = H we get: Corollary 4.5 We can easily show that: Corollary 4.6 ∆(G ∗ G; x) = ∆(G; x) 2 − ∆(G; x) 1 1 1 1 − 1= − 1+ − 1+... + − 1 ∆(G1 ∗ G2 ∗ ... ∗ Gk; x) ∆(G1; x) ∆(G2; ... |

2 |
Distance Degree Regular Graphs.” The Theory and Application of Graphs
- Bloom, Kennedy, et al.
- 1981
(Show Context)
Citation Context ...generating function for γ(G, v, n) ofG at v. Γ(G, v; x) = ∞� γ(G, v, n)x n . n=0 Note that the spherical growth sequence {δ(G, v, n)|n =0, 1, 2, ...} is sometimes called the distance degree sequence, =-=[2, 3, 11]-=-. Let D(G) ={∆(G, v; x)|v from V (G)} and G(G) ={Γ(G, v; x)|v from V (G)} In finite graphs define the averages 156s¯∆(G; x) = 1 |V | � v ∆(G, v; x) and ¯ Γ(G; x) = 1 |V | � Γ(G, v; x) Instead of δ(G, ... |

2 |
Distance degrees of vertex-transitive graphs
- Hilano
- 1989
(Show Context)
Citation Context ...generating function for γ(G, v, n) ofG at v. Γ(G, v; x) = ∞� γ(G, v, n)x n . n=0 Note that the spherical growth sequence {δ(G, v, n)|n =0, 1, 2, ...} is sometimes called the distance degree sequence, =-=[2, 3, 11]-=-. Let D(G) ={∆(G, v; x)|v from V (G)} and G(G) ={Γ(G, v; x)|v from V (G)} In finite graphs define the averages 156s¯∆(G; x) = 1 |V | � v ∆(G, v; x) and ¯ Γ(G; x) = 1 |V | � Γ(G, v; x) Instead of δ(G, ... |

2 | The growth function of a graph group - Lewin - 1989 |

2 | The growth function of some free products of groups - Lewin - 1991 |

2 | Notes for a course in Rome, Chapters 5 and 6, Serie formali e gra di Cayley, and Funzioni di crescita per i gruppi, manuscript - Machi - 1997 |

2 | Distance degree regular graphs - Bloom, Quintas, et al. - 1980 |

1 |
Some problems concerning distance and path degree sequences
- Bloom, Kennedy, et al.
- 1981
(Show Context)
Citation Context ...generating function for γ(G, v, n) ofG at v. Γ(G, v; x) = ∞� γ(G, v, n)x n . n=0 Note that the spherical growth sequence {δ(G, v, n)|n =0, 1, 2, ...} is sometimes called the distance degree sequence, =-=[2, 3, 11]-=-. Let D(G) ={∆(G, v; x)|v from V (G)} and G(G) ={Γ(G, v; x)|v from V (G)} In finite graphs define the averages 156s¯∆(G; x) = 1 |V | � v ∆(G, v; x) and ¯ Γ(G; x) = 1 |V | � Γ(G, v; x) Instead of δ(G, ... |

1 |
Growth in repeated truncations of maps, Atti
- Pisanski, Tucker
(Show Context)
Citation Context ... Section 6 we consider the lexicographic product of graphs, we also mention tensor product and some other operations on graphs. One such operation is truncation and was considered quite recently; see =-=[18]-=-. It may come as a surprise but the growth function of the tensor product is not determined by the growth functions of its factors. 2 Growthinrootedgraphs Let G be connected, finite or locally finite ... |

1 | P de la Harpe, On Finitely Generated Groups and Problems related to Growth, manuscript - Grigorchuk |

1 | RComp: A Mahtematica Package for Computing with Recursive Sequences - Nemes, Petkovsek - 1995 |