Dr. Shen received his Ph.D. from Queens University, Canada in 1998. He has been working on combinatorial matrix theory, a subject which connects linear algebra with combinatorics. He has proved a few conjectures on the exponents of primitive matrices. Dr. Shen's current research is focused on graph theory and algorithm. He has made substantial contributions to the research of a number of conjectures, including the Caccetta-Haggkuist conjecture (1978), on the cycle structure of directed graphs. He proved the up-to-date strongest result to support the Frame-Stewart conjecture (1941) on the towers of Hanoi. Dr. Shen plans to extend his research to networks design and probabilistic methods in discrete mathematics.

Selected Publications:

• J. Shen, B. Liu and X. Wang; On the largest eigenvalue of non-regular graphs, J. Combin. Theory, Ser. B. 97(6):1010-1018 (2007).
• G. Chen, J. Hutchinson, K. Keating, and J. Shen. Characterization of [1,k]-Bar Visibility Trees. Electronic J. Combin. 13, Research Paper 90, 12 pages, 2006.
• J. Bondy, J. Shen, S. Thomasse, and C. Thomassen. Density conditions for triangles in multipartite graphs. Combinatorica, 26(2):121-131, 2006.
• J. Shen. On two Turan numbers. J. Graph Theory, 51(3):244-250, 2006.
• X. Chen and J. Shen. Improved schemes for power-efficient broadcast in Ad Hoc networks. International Journal of High Performance Computing and Networking, 4(2) Research paper 9, 2006.
• L. You, B. Liu, and J. Shen. r-indecomposable and r-nearly decomposable matrices. Linear Algebra Appl. 407:105-116, 2005.
• X. Chen and J. Shen. Maximum possible hop count for package routing in MANET, in: Handbook on Theoretical and Algorithmic Aspects of Sensor, Ad Hoc Wireless and Peer-to-Peer Networks, Auerbach Publications, pp. 43-52, 2005.
• X. Chen and J. Shen. On the Frame-Stewart conjecture about the towers of Hanoi. SIAM J. Computing, 33(3):584-589, 2004.
• J. Shen, L. Sheng, and J. Wu. Searching for sorted sequences of kings in tournaments. SIAM J. Computing, 32(5):1201-1209, 2003.
• J. Shen. Short cycles in digraphs with local average outdegree at least two. Electronic J. Combin. 10(1) Research paper 26, 11 pages. (electronic), 2003.
• G. Chen, J. Shen, and R. Yuster. Second neighborhood via first neighborhood in digraphs, Ann. Comb. 7(1):15-20, 2003.
• J. Shen and C. J. Wyels. On the number of arcs in primitive digraphs with large exponents. Linear Algebra Appl. 364:243-251, 2003.
• B. Zhou and J. Shen. On generalized exponents of tournaments. Taiwanese J. Math. 6(4):565-572, 2002.
• J. Shen. On the Caccetta-Haggkvist conjecture. Graphs Combin. 18(3):645-654, 2002.
• R. Brualdi and J. Shen. Disjoint cycles in Eulerian digraphs and the diameter of interchange graphs. J. Combin. Theory B, 85(2):189-196, 2002.
• J. Shen and R. Yuster. A note on the number of edges guaranteeing a C4 in Eulerian bipartite digraphs. Electronic J. Combin. 9(1), Note 6, 6 pages. (electronic), 2002.
• R. Brualdi and J. Shen. Landau's inequalities for tournament scores and a short proof of a theorem on transitive sub-tournaments. J. Graph Theory, 38(4):244-254, 2001.
• D. Gregory and J. Shen. On the girth of digraphs with high connectivity. Ars Combin. 60:201-207, 2001.
• R. A. Brualdi and J. Shen. Diameter of the NEPS of bipartite graphs. Discrete Math., 226(1-3):373--376, 2001.
• B. Liu, Z. Bo, Q. Li, and J. Shen. Generalized index of Boolean matrices. Ars Combin., 57:247--255, 2000.
• J. Shen and D. Gregory. Exponents of vertex-transitive digraphs. Discrete Math. 212(3):245-255, 2000.
• J. Shen. Exponents of 2-regular digraphs. Discrete Math., 214(1-3):211--219, 2000.
• J. Shen. On the girth of digraphs. Discrete Math., 211(1-3):167--181, 2000.
• J. Shen, D. Gregory, and S. Neufeld. Exponents of indecomposability. Linear Algebra Appl., 288(1-3):229--241, 1999.
• R. A. Brualdi and J. Shen. Discrepancy of matrices of zeros and ones. Electron. J. Combin., 6(1): Research Paper 15, 12 pages. (electronic), 1999.
• S. Neufeld and J. Shen. Some results on generalized exponents. J. Graph Theory, 28(4):215--225, 1998.
• J. Shen. Directed triangles in digraphs. J. Combin. Theory Ser. B, 74(2):405--407, 1998.
• J. Shen and S. Neufeld. Local exponents of primitive digraphs. Linear Algebra Appl., 268:117--129, 1998.
• J. Shen and S. Neufeld. On a problem of Lewin. Linear Algebra Appl., 274:411--426, 1998.
• J. Xu, J. Shen, and Z. Li. Some results on arithmetic and balanced graphs. J. China Univ. Sci. Tech., 28(2):183--193, 1998.
• J. Shen. A bound on the exponent of primitivity in terms of diameter. Linear Algebra Appl., 244:21--33, 1996.
• J. Shen. An improvement of the Dulmage-Mendelsohn theorem. Discrete Math., 158(1-3):295--297, 1996.
• J. Shen. A problem on the exponent of primitive digraphs. Linear Algebra Appl., 244:255--264, 1996.
• J. Shen. Some estimated formulas for the Frobenius numbers. Linear Algebra Appl., 244:13--20, 1996.
• J. Shen and D. A. Gregory. Some eigenvalue inequalities for matrices with restricted entries. Linear and Multilinear Algebra, 41(2):107--112, 1996.
• J. Shen. Proof of a conjecture about the exponent of primitive matrices. Linear Algebra Appl., 216:185--203, 1995.
• J. Shen and Q. Li. Two theorems on double loop networks. J. China Univ. Sci. Tech., 25(2):127--132, 1995.