A <i>q</i>-Analogue of the Addressing Problem of Graphs by Graham and Pollak

Watanabe, Satoshi; Ishii, Kota; Sawa, Masanori

doi:10.1137/110831520

Cited by 12 publications

(11 citation statements)

References 10 publications

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“…Watanabe, Ishii and Sawa [14] studied the optimal p0, 1, 2, ˚q-addressings of various graphs. They observed the following pattern for odd cycles N 3 pC 5 q " 3, N 3 pC 7 q " 4, N 3 pC 9 q " 5 and asked the natural question whether N 3 pC 2n`1 q " n `1 for n ě 5 ?…”

Section: Odd Cyclesmentioning

confidence: 99%

“…The best known lower bound is N 2 pJpn, kqq ě n (see [4,Theorem 5.3]). For r ě 3, Watanabe, Ishii and Sawa [14] studied p0, 1, . .…”

Section: Introductionmentioning

confidence: 99%

“…Note that the stronger result N r pGq ě maxpn `pDq, n ´pDq{pr ´1qq follows from the work of Gregory and Vander Meulen [9, Theorem 4.1] (see also [13]). In [14], the first three authors prove that the Petersen graph can be optimally addressed with p0, 1, 2, ˚q-words of length 4 and show that N r pC n q " n{2 for any n even and any r ě 3. For odd cycles, they prove that N 3 pC 2n`1 q " n `1 for n P t2, 3, 4u and ask whether this statement is true for larger values of n. In this paper, we determine that this is true for n " 5 and N 3 pC 11 q " 6, but fails for n P t6, 7, 8, 9u, where N 3 pC 13 q " 8, N 3 pC 15 q " 9, N 3 pC 17 q " 10 and N 3 pC 19 q " 11.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Addressing Johnson graphs, complete multipartite graphs, odd cycles and other graphs

Alon¹,

Cioabă²,

Gilbert³

et al. 2018

Preprint

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Graham and Pollak showed that the vertices of any graph G can be addressed with N -tuples of three symbols, such that the distance between any two vertices may be easily determined from their addresses. An addressing is optimal if its length N is minimum possible.In this paper, we determine an addressing of length kpn ´kq for the Johnson graphs Jpn, kq and we show that our addressing is optimal when k " 1 or when k " 2, n " 4, 5, 6, but not when n " 6 and k " 3. We study the addressing problem as well as a variation of it in which the alphabet used has more than three symbols, for other graphs such as complete multipartite graphs and odd cycles. We also present computations describing the distribution of the minimum length of addressings for connected graphs with up to 10 vertices. Motivated by these computations we settle a problem of Graham, showing that most graphs on n vertices have an addressing of length at most n ´p2 ´op1qq log 2 n.

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Section: Odd Cyclesmentioning

confidence: 99%

“…The best known lower bound is N 2 pJpn, kqq ě n (see [4,Theorem 5.3]). For r ě 3, Watanabe, Ishii and Sawa [14] studied p0, 1, . .…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Addressing Johnson graphs, complete multipartite graphs, odd cycles and other graphs

Alon¹,

Cioabă²,

Gilbert³

et al. 2018

Preprint

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show abstract

“…Graham and Pollak [13,14] also determined N (K n,m ) for many values of n and m. The determination of N (K n,m ) for all values of n and m was completed by Fujii and Sawa [11]. A more general addressing scheme, allowing the addresses to contain more than two different nonzero symbols, was recently studied by Watanabe, Ishii and Sawa [23]. The parameter N (G) has been determined when G is a tree or a cycle [14], as well as one particular triangular graph T 4 [25], described in Section 5.…”

Section: Graph Addressingsmentioning

confidence: 99%

Addressing graph products and distance-regular graphs

Cioabă

Elzinga

Markiewitz

et al. 2017

Discrete Applied Mathematics

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Graham and Pollak showed that the vertices of any connected graph G can be assigned t-tuples with entries in {0, a, b}, called addresses, such that the distance in G between any two vertices equals the number of positions in their addresses where one of the addresses equals a and the other equals b. In this paper, we are interested in determining the minimum value of such t for various families of graphs. We develop two ways to obtain this value for the Hamming graphs and present a lower bound for the triangular graphs.

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“…al. [17] studied a q-ary extension of the classical binary addressing problem of graphs which was originally posed by Graham and Pollak [5], and found a sharp lower bound for the minimum length of addressings in terms of distance eigenvalues of uniform hypertrees. Lin and Zhou [8] and Lin et al [10] studied the distance spectral radius of uniform hypergraphs and particularly, uniform hypertrees.…”

mentioning

confidence: 99%

Extremal properties of the distance spectral radius of hypergraphs

Wang¹,

Zhou

2020

ELA

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The distance spectral radius of a connected hypergraph is the largest eigenvalue of its distance matrix. The unique hypertrees with minimum distance spectral radii are determined in the class of hypertrees of given diameter, in the class of hypertrees of given matching number, and in the class of non-hyperstar-like hypertrees, respectively. The unique hypergraphs with minimum and second minimum distance spectral radii are determined in the class of unicylic hypergraphs. The unique hypertree with maximum distance spectral radius is determined in the class of $k$-th power hypertrees of given matching number.

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A q-Analogue of the Addressing Problem of Graphs by Graham and Pollak

Cited by 12 publications

References 10 publications

Addressing Johnson graphs, complete multipartite graphs, odd cycles and other graphs

Addressing Johnson graphs, complete multipartite graphs, odd cycles and other graphs

Addressing graph products and distance-regular graphs

Extremal properties of the distance spectral radius of hypergraphs

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