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**Date:** 2011
**Keyword:** strong vertex magic total labellings | even regular odd degree graphs | Gray's construction
**Language:** eng
**Resource Type:** report
**Identifier:** uon:7667
**Description:** Tables of edge cycles proving the theorem: Other than the three graphs, $C_{3} cup C_{4}$, $2C_{3} cup C_{5}$ and $3C_{3} cup C_{4}$, every 2-regular graph of odd order less than 30 has a strong verte... More
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**Date:** 2008
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.13/41712
**Description:** We generalise the notion of Heron triangles to rational-sided, cyclic n-gons with rational area using Brahmagupta's formula for the area of a cyclic quadrilateral and Robbins' formulæ for the area of ... More
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**Date:** 2005
**Language:** eng
**Resource Type:** journal article
**Identifier:** uon:122
**Description:** This paper discusses tetrahedra with rational edges forming an arithmetic progression, focussing specifically on whether they can have rational volume or rational face areas. Several infinite families... More
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**Date:** 2008
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.13/42850
**Description:** This paper discusses tetrahedra with rational edges forming a geometric progression, focussing on whether they can have rational volume or rational face areas. We examine the 30 possible configuration... More
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**Date:** 2006
**Language:** eng
**Resource Type:** journal article
**Identifier:** uon:6510
**Description:** The level set G(n,m) comprises all unlabelled simple graphs of order n and size m, and is partitioned into similarity classes, comprising all graphs with the same degree sequence. When graphs are orde... More
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**Date:** 2006
**Resource Type:** journal article
**Identifier:** uon:1082
**Description:** Buchholz [R.H. Buchholz, Perfect pyramids, Bull. Austral. Math. Soc. 45 (1991) 353–368] began a systematic search for tetrahedra having integer edges and volume by restricting his attention to those w... More
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**Date:** 1999
**Keyword:** number theory | 51M04 | 11D25 | diophantine equations | cubic and quartic equations | elementary problems in Euclidean geometries
**Language:** eng
**Resource Type:** journal article
**Identifier:** uon:6511
**Description:** We study triangles and cyclic quadrilaterals which have rational area and whose sides form geometric or arithmetic progressions. A complete characterization is given for the infinite family of triangl... More
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**Date:** 2009
**Language:** eng
**Resource Type:** report
**Identifier:** http://hdl.handle.net/1959.13/38076
**Description:** A Chinese Remainder method is developed for determining likely positive integers n starting maximal runs of size s and constant principal rank r, that is, {n + i ∈ ℤ⁺, 0 ≤ i < s : ω(n + i) = r} with ω... More
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**Date:** 2009
**Language:** eng
**Resource Type:** report
**Identifier:** http://hdl.handle.net/1959.13/35886
**Description:** The principal divisors of a positive integer n are its maximal prime-power divisors. The principal divisor rank ω(n) is the number of such divisors, also equal to the number of distinct prime divisors... More
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**Date:** 2009
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.13/916345
**Description:** A graph H of order h is an n-linked cycle if it has an induced subgraph G of order g < h and an automorphism α: H → H of order n ≥ 2 such that H = ⋃{αr(G) : 0 ≤ r < n} and G has an induced subgraph K ... More
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**Date:** 2009
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.13/916332
**Description:** A linked pair is a graph H = G₀ ⋃ G₁ formed from (1) a given finite graph G,(2) isomorphic induced proper subgraphs K and K*, not necessarily distinct, and (3) a graph isomorphism σ: K* → K. The graph... More
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**Date:** 2006
**Language:** eng
**Resource Type:** journal article
**Identifier:** uon:6466
**Description:** A sparse anti-magic square is an n X n array whose non-zero entries are the consecutive integers 1,...,m for some m ≤ n² and whose row-sums and column-sums form a set of consecutive integers: We deriv... More
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**Date:** 2006
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.13/26848
**Description:** We introduce a generalisation of the traditional magic square which proves useful in the construction of magic labelings of graphs. An order n sparse semi-magic square is an x n array containing the e... More
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**Date:** 2009
**Keyword:** magic labeling
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.13/915980
**Description:** Previously the first author has shown how to construct vertex-magic total labelings (VMTLs) for large families of regular graphs. The construction proceeds by successively adding arbitrary 2-factors t... More
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**Date:** 2012
**Language:** eng
**Resource Type:** journal article
**Identifier:** http://hdl.handle.net/1959.13/933758
**Description:** We survey and update data on runs of consecutive integers each having exactly r distinct prime factors (briefly, of principal rank r). For 3 ≤ r ≤ 64 and other sporadic values, lower bounds are given ... More
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