### Abstract

Let N denote the set of positive integers and Z denote all integers. The (integral) sum graph of a finite subset S ⊂ N(Z) is the graph (S, E) with uv ∈ E if and only if u + v∈S. A graph G is said to be an (integral) sum graph if it is isomorphic to the (integral) sum graph of some S ⊂= N(Z). The (integral) sum number of a given graph G is the smallest number of isolated nodes which when added to G result in an (integral) sum graph. We show that the integral sum number of a complete graph with n ≥ 4 nodes equals 2n - 3, which proves a conjecture of Harary. And we disprove another conjecture of Harary by showing that there are infinitely many trees which are not caterpillars but are integral sum graphs.

Original language | English (US) |
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Pages (from-to) | 241-244 |

Number of pages | 4 |

Journal | Discrete Mathematics |

Volume | 160 |

Issue number | 1-3 |

DOIs | |

State | Published - Nov 15 1996 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

*Discrete Mathematics*,

*160*(1-3), 241-244. https://doi.org/10.1016/0012-365X(95)00163-Q

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*Discrete Mathematics*, vol. 160, no. 1-3, pp. 241-244. https://doi.org/10.1016/0012-365X(95)00163-Q

**Harary's conjectures on integral sum graphs.** / Chen, Zhibo.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Harary's conjectures on integral sum graphs

AU - Chen, Zhibo

PY - 1996/11/15

Y1 - 1996/11/15

N2 - Let N denote the set of positive integers and Z denote all integers. The (integral) sum graph of a finite subset S ⊂ N(Z) is the graph (S, E) with uv ∈ E if and only if u + v∈S. A graph G is said to be an (integral) sum graph if it is isomorphic to the (integral) sum graph of some S ⊂= N(Z). The (integral) sum number of a given graph G is the smallest number of isolated nodes which when added to G result in an (integral) sum graph. We show that the integral sum number of a complete graph with n ≥ 4 nodes equals 2n - 3, which proves a conjecture of Harary. And we disprove another conjecture of Harary by showing that there are infinitely many trees which are not caterpillars but are integral sum graphs.

AB - Let N denote the set of positive integers and Z denote all integers. The (integral) sum graph of a finite subset S ⊂ N(Z) is the graph (S, E) with uv ∈ E if and only if u + v∈S. A graph G is said to be an (integral) sum graph if it is isomorphic to the (integral) sum graph of some S ⊂= N(Z). The (integral) sum number of a given graph G is the smallest number of isolated nodes which when added to G result in an (integral) sum graph. We show that the integral sum number of a complete graph with n ≥ 4 nodes equals 2n - 3, which proves a conjecture of Harary. And we disprove another conjecture of Harary by showing that there are infinitely many trees which are not caterpillars but are integral sum graphs.

UR - http://www.scopus.com/inward/record.url?scp=0042187901&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0042187901&partnerID=8YFLogxK

U2 - 10.1016/0012-365X(95)00163-Q

DO - 10.1016/0012-365X(95)00163-Q

M3 - Article

AN - SCOPUS:0042187901

VL - 160

SP - 241

EP - 244

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1-3

ER -