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Title: Mathematical foundations of the GraphBLAS

The GraphBLAS standard ( is being developed to bring the potential of matrix-based graph algorithms to the broadest possible audience. Mathematically, the GraphBLAS defines a core set of matrix-based graph operations that can be used to implement a wide class of graph algorithms in a wide range of programming environments. This study provides an introduction to the mathematics of the GraphBLAS. Graphs represent connections between vertices with edges. Matrices can represent a wide range of graphs using adjacency matrices or incidence matrices. Adjacency matrices are often easier to analyze while incidence matrices are often better for representing data. Fortunately, the two are easily connected by matrix multiplication. A key feature of matrix mathematics is that a very small number of matrix operations can be used to manipulate a very wide range of graphs. This composability of a small number of operations is the foundation of the GraphBLAS. A standard such as the GraphBLAS can only be effective if it has low performance overhead. Finally, performance measurements of prototype GraphBLAS implementations indicate that the overhead is low.
 [1] ;  [2] ;  [3] ;  [4] ;  [5] ;  [6] ;  [7] ;  [8] ;  [2] ;  [9] ;  [10] ;  [8] ;  [11] ;  [11] ;  [2] ;  [12]
  1. MIT Lincoln Lab. Supercomputing Center, Lexington, MA (United States)
  2. Indiana Univ., Bloomington, IN (United States)
  3. Georgia Inst. of Technology, Atlanta, GA (United States)
  4. Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
  5. Carnegie Mellon Univ., Pittsburgh, PA (United States)
  6. Univ. of California, Santa Barbara, CA (United States)
  7. Univ. of Washington, Seattle, WA (United States)
  8. IBM, Armonk, NY (United States)
  9. Karlsruhe Inst. of Technology (KIT) (Germany)
  10. CMU Software Engineering Inst., Pittsburgh, PA (United States)
  11. Univ. of California, Davis, CA (United States)
  12. Intel, Santa Clara, CA (United States)
Publication Date:
Grant/Contract Number:
AC02-05CH11231; DMS-1312831; FA8721-05-C-0003
Accepted Manuscript
Journal Name:
2016 IEEE High Performance Extreme Computing Conference, HPEC 2016
Additional Journal Information:
Journal Name: 2016 IEEE High Performance Extreme Computing Conference, HPEC 2016
Research Org:
Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
Sponsoring Org:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21); National Science Foundation (NSF); USDOD
Country of Publication:
United States
97 MATHEMATICS AND COMPUTING; matrices; sparse matrices; finite element analysis; standards; additives; programming environments; graph theory; mathematics computing; matrix algebra
OSTI Identifier: