### Abstract

We introduce a new graph invariant Λ(G) that we call maxmaxflow, and put it in the context of some other well-known graph invariants, notably maximum degree and its relatives. We prove the equivalence of two "dual" definitions of maxmaxflow: one in terms of flows, the other in terms of cocycle bases. We then show how to bound the total number (or more generally, total weight) of various classes of subgraphs of G in terms of either maximum degree or maxmaxflow. Our results are motivated by a conjecture that the modulus of the roots of the chromatic polynomial of G can be bounded above by a function of Λ(G).

Original language | English (US) |
---|---|

Pages (from-to) | 1-46 |

Number of pages | 46 |

Journal | Electronic Journal of Combinatorics |

Volume | 17 |

Issue number | 1 |

State | Published - 2010 |

### Fingerprint

### Keywords

- Chromatic polynomial
- Cocycle
- Degeneracy number
- Flow
- Graph
- Maximum degree
- Maxmaxflow
- Second-largest degree
- Subgraph

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Geometry and Topology
- Theoretical Computer Science

### Cite this

*Electronic Journal of Combinatorics*,

*17*(1), 1-46.

**Maxmaxflow and counting subgraphs.** / Jackson, Bill; Sokal, Alan D.

Research output: Contribution to journal › Article

*Electronic Journal of Combinatorics*, vol. 17, no. 1, pp. 1-46.

}

TY - JOUR

T1 - Maxmaxflow and counting subgraphs

AU - Jackson, Bill

AU - Sokal, Alan D.

PY - 2010

Y1 - 2010

N2 - We introduce a new graph invariant Λ(G) that we call maxmaxflow, and put it in the context of some other well-known graph invariants, notably maximum degree and its relatives. We prove the equivalence of two "dual" definitions of maxmaxflow: one in terms of flows, the other in terms of cocycle bases. We then show how to bound the total number (or more generally, total weight) of various classes of subgraphs of G in terms of either maximum degree or maxmaxflow. Our results are motivated by a conjecture that the modulus of the roots of the chromatic polynomial of G can be bounded above by a function of Λ(G).

AB - We introduce a new graph invariant Λ(G) that we call maxmaxflow, and put it in the context of some other well-known graph invariants, notably maximum degree and its relatives. We prove the equivalence of two "dual" definitions of maxmaxflow: one in terms of flows, the other in terms of cocycle bases. We then show how to bound the total number (or more generally, total weight) of various classes of subgraphs of G in terms of either maximum degree or maxmaxflow. Our results are motivated by a conjecture that the modulus of the roots of the chromatic polynomial of G can be bounded above by a function of Λ(G).

KW - Chromatic polynomial

KW - Cocycle

KW - Degeneracy number

KW - Flow

KW - Graph

KW - Maximum degree

KW - Maxmaxflow

KW - Second-largest degree

KW - Subgraph

UR - http://www.scopus.com/inward/record.url?scp=77955634451&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77955634451&partnerID=8YFLogxK

M3 - Article

VL - 17

SP - 1

EP - 46

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 1

ER -