### Table 1. Properties of the test graphs

2005

"... In PAGE 6: ... Our test set consists of 19 graphs obtained from molecular dynamics and finite element applications [8, 13]. Table1 displays the structural properties of the test graphs, including maximum, mini- mum, and average degree. The table also displays the number of colors and the runtime in seconds used by a sequential FF algorithm when run on a single node of our test plat- form.... In PAGE 6: ... The table also displays the number of colors and the runtime in seconds used by a sequential FF algorithm when run on a single node of our test plat- form. All of the results presented in this section are average performance results over all of the graphs presented in Table1 . Each individual test is an average of 5 runs.... In PAGE 9: ... The core algorithm is a nontrivial way of coloring the boundary vertices in parallel. Figure 4(a) shows the percentage of boundary vertices for the graphs in Table1 when using block partitioning with the natural vertex order- ing, and when using Metis. As one can see the number of boundary vertices increases with the number of processors being used.... In PAGE 10: ...4. (a) Percentage of boundary vertices for graphs in Table1 (N = natural ordering, V = ordering given by Metis), and random graphs. (b) Speedup for random graphs of various average degrees 5Conclusion We have developed an efficient and truly scalable parallel graph coloring algorithm suitable for a distributed memory computer.... ..."

Cited by 8

### Table 1. Properties of the test graphs.

2005

"... In PAGE 5: ... Our test set consists of 19 graphs obtained from molecular dynamics and finite element applications [8, 13]. Table1 displays the structural properties of the test graphs, including maximum, mini- mum, and average degree. The table also displays the number of colors and the runtime in seconds used by a sequential FF algorithm when run on a single node of our test plat- form.... In PAGE 5: ... The table also displays the number of colors and the runtime in seconds used by a sequential FF algorithm when run on a single node of our test plat- form. All of the results presented in this section are average performance results over all of the graphs presented in Table1 . Each individual test is an average of 5 runs.... In PAGE 9: ... boundary vertices in parallel. Figure 4(a) shows the percentage of boundary vertices for the graphs in Table1 when using block partitioning with the natural vertex ordering (blue), and when using Metis (red). As one can see the number of boundary vertices increases with the number of processors being used.... In PAGE 9: ... 4. (a) Percentage of boundary vertices for graphs in Table1 (blue and red), and... ..."

Cited by 8

### Table 1: Asymptotic degree-diameter properties of the difierent graphs.

in Graph-Theoretic Analysis of Structured Peer-to-Peer Systems: Routing Distances and Fault Resilience

2003

"... In PAGE 4: ... As we show in section 7, distributed de Bruijn graphs possess no more conceptual complexity than Chord, achieve optimal diameter in the peer-to-peer graph, and can be built with a flxed application-layer degree. Table1 shows asymptotic diameter and node degree of de Bruijn graphs and several existing (deterministic) struc- tures. First note that we assume that CAN uses circular (toroidal) routing in each of the dimensions, which means that all nodes along the borders maintain 2d neighbors and that the CAN graph is regular.... ..."

Cited by 68

### Table 2: The order of the largest known graphs of maximum degree and diameter D.

2005

"... In PAGE 16: ...Table2 shows a summary of current largest known graphs for degree 16 and diameter D 10. These graphs provide the best current lower bounds on the order of graphs for given values of degree and diameter.... In PAGE 16: ...fr quot;. Recent updates in Table2 are due to Exoo: entries (3,6)-(3,8), (4,4), (4,7), (5,3), (5,5), (6,3), (6,4), (7,3), (16,2); to Hafner: entries (5,9), (5,10), (6,7)-(6,10), (7,6)-(7,10), (8,5), (8,7), (8,9), (8,10), (9,7), (9,10), (10,5), (10,7)-(10,10), (11,5), (11,7), (11,8), (12,7), (13,5), (13,7), (13,8), (14,5), (14,8), (15,8); to Quisquater: entries (3,9), (3,10); to G omez and Pelayo: entries (5,6), (6,6), (8,6), (9,6), (10,6), (12,6), (14,9); to Sampels: entries (4,8), (4,10), (5,8)-(5,10), (6,7)-(6,10), (7,6)-(7,10), (8,8)-(8,10), (9,4), (9,5), (9,8)-(9,10), (10,5), (10,7), (10,8)-(10,10); to McKay, Miller, Sir a n: entries (11,2), (13,2); and to G omez: entries (5,6), (8,6), (9,6), (10,6), (12,6), (14,6) [89]. 2.... In PAGE 18: ... The survey emphasises algebraic features, such as cosets, conjugacy classes, and automorphism actions, in the determination of some topological properties of over 18 types of networks. We note that roughly one half of the values in Table2 have been obtained from Cayley graphs. Computer-assisted constructions of large ( ;D)-graphs, for relatively small and D, from Cayley graphs of semidirect products of (mostly cyclic) groups can be found in Hafner [155].... ..."

Cited by 7

### Table 1: Bounds on the complexity of Property Testing of G-isomorphism.

2007

"... In PAGE 3: ... The query-2 complexity of 1-sided and 2-sided error G-isomorphism testing is O(pn(1 + log jGj)). In Table1 , we abbreviated the expression 1 + log jGj to log jGj for better typography. The only case where this makes a difference is when jGj = 1 so the results as stated in the Table 1 assume jGj 2.... In PAGE 3: .... The query-2 complexity of 1-sided and 2-sided error G-isomorphism testing is O(pn(1 + log jGj)). In Table 1, we abbreviated the expression 1 + log jGj to log jGj for better typography. The only case where this makes a difference is when jGj = 1 so the results as stated in the Table1 assume jGj 2. Theorem 1.... In PAGE 3: ...1). Table1 summarizes our results on G-isomorphism. Table 2 gives the results of Fischer and Matsliah on Graph Isomorphism.... ..."

Cited by 1

### Table 1. The set of test graphs.

"... In PAGE 5: ... The quotient graphs were obtained using Unbalanced Recursive Bisection (URB) in Par2 [11]. Table1 summarizes the test suite of graphs and their properties. In the table, 1 and n?1 are the largest and sec- ond smallest eigenvalues of the Laplacian matrix L = D ? A associated with graph G (here A is the adjacency matrix of G and D is the diagonal matrix of the degrees of the nodes).... In PAGE 7: ... In fact, for regular graphs, 1(L) = 2d, and, usually, for irregular graphs, 1(L) 2d. For the graphs in our test suite ( Table1 ), we observed that b 1(L) d + d, where d is the average degree of the graph, and we used this as an estimate of 1(L). Estimating n?1(L): Our approach to estimate n?1(L) is via the isoperimetric constant of the graph.... In PAGE 7: ... Our experiments on the test suite (cf. Table1 ) suggest that 12h0(G) is a good estimation for n?1. In summary, we use b 1 = d + d and b n?1 = 12h0(G), and choose b and b as... ..."

### Table 2 presents the compression results in terms of total bits required divided by edges in the graph. For the random graphs, we have taken the average of ten trials, where a different random graph is produced for each trial. We note that there is little deviation between the runs. For the uncompressed size in bits per edge, we use the underestimate log2 (#nodes). As seen in graph G1, when the amount of copying is low, and thus the average degree is very small, the reference algorithm alone does slightly worse than the Huffman algorithm, although using a Huffman code in conjunction with the reference algorithm leads to better performance.

2001

"... In PAGE 9: ...20 8.35 Table2 : Results from the test graphs; bits per edge. 6 Future Work We have initiated study into how to compress Web graphs using the copy graph model, a random graph family with properties similar to Web graphs.... ..."

Cited by 49

### Table 2 presents the compression results in terms of total bits required divided by edges in the graph. For the random graphs, we have taken the average of ten trials, where a different random graph is produced for each trial. We note that there is little deviation between the runs. For the uncompressed size in bits per edge, we use the underestimate D0D3CVBE (#nodes). As seen in graph BZBD, when the amount of copying is low, and thus the average degree is very small, the reference algorithm alone does slightly worse than the Huffman algorithm, although using a Huffman code in conjunction with the reference algorithm leads to better performance.

2001

"... In PAGE 9: ...20 8.35 Table2 : Results from the test graphs; bits per edge. 6 Future Work We have initiated study into how to compress Web graphs using the copy graph model, a random graph family with properties similar to Web graphs.... ..."

Cited by 49

### Table 1: A summary of the test graphs size degree

"... In PAGE 5: ... The test suite also includes one non mesh-based graph, add32. Table1 gives a list of the graphs, their sizes, the maximum, minimum amp; average degree of the vertices and a short de- scription. The degree information (the degree of a vertex is the number of vertices adjacent to it) gives some idea of the character of the graphs.... ..."

### Table 3: Performance under random failures: ran- dom graph with 105 nodes, avg. degree of 7. The an- alytic bound predicts the number of replicas which in this case is equal to the number of probes.

"... In PAGE 8: ...5), we consider how the system can adapt at run-time as it becomes aware of new failures. In Table3 , we consider a random graph with 100,000... ..."