Professor, Mathematics, School of Engineering, Coimbatore

Qualification:

Ph.D, MPhil, MSc

Email:

s_sundaram(at)cb.amrita.edu

Phone:

+91- 422 – 2685600

Dr. Somasundaram K. currently serves as Professor in the Department of Mathematics, School of Engineering, Coimbatore Campus. His areas of research include Multilinear Algebra, Graph Theory, Graph Theory Algorithms in Network on Chip.

**Ph.D. in Mathematics**

Bharathiyar University, Coimbatore, India, 2005

Dissertation: Permanents: Mates, Stars and Graphs.

**Post Doctorial Study**

Department of Computer Science, Turku University, Turku, Finland, 2009.

- Multi-Linear Algebra, in particular Permanents.
- Graph Coloring, in particular Total Coloring.
- Graph Theory Algorithms in Network on Chip (NoC).

- Dr. Dharmarajan – Graph Theory

- Dr. P. Subramanian - Completed (Conjectures on Permanents)
- Dr. J. Geetha - Completed (Total coloring conjecture)
- Dr. S. Mohan - Completed (Total coloring conjecture)
- Mr. R.Vignesh - on going (Total coloring conjecture)
- Ms. T G Thurshara - Completed (Static Analysis)
- Ms. Hemalatha - on going (Graph Theory)
- Ms. Sandhya - on going (Graph Theory)
- Ms. K Divya - on going (Conjectures on Permanents)

- Amrita Vishwa Vidya peetham Grant (AVVP/05/KSS/Maths /C/IFRP/09) “ Investigation of a Graph Partitioning Algorithm for VLSI Circuits” awarded January 2006.
- NBHM Grant (2/48(7)/2013/NBHM(R.P)/R&D II /1535), “Total Coloring Conjecture for Certain Classes of Graphs” 2014.
- DST Grant (DST/SERB No: SR/S4/MS : 867/14), “Behzad-Vizing Conjecture on Graph Coloring for Product Graphs” 2015.
- SERB Grand (EMR/2017/001869), Total Colorings for Certain Classes of Cayley Graphs, 2018.

- Organising Secretary: “
**International Conference on Graph Theory and Its Applications**” during December 11-13, 2008. - Organising Secretary: “
**Second India-Taiwan Conference on Discrete Mathematics**” during September 8-11, 2011. - Organising Secretary: “
**International Conference on Graph Theory and Its Applications**” during December 17-21, 2015. - Organising Secretary: “
**Workshop on Interconnection Networks**” during April 24-28, 2017. - Organising Secretary: “
**International Conference on Graph Theory and Its Applications**” during January 4-6, 2019.

Year of Publication | Title |
---|---|

2021 |
C. Gopika, J. Geetha, and Dr. Somasundaram K., “Weighted PI index of tensor product and strong product of graphs”, Discrete Mathematics, Algorithms and Applications, vol. 13, p. 2150019, 2021.[Abstract] The Weighted Padmakar–Ivan (PI) index of a connected, simple graph G is given by PIw(G) =∑e=(u,v)∈E(G)((dG(u) + dG(v))(|V (G)|− NG(e))), where NG(e) denotes the number of equidistant vertices of the edge e. In this paper, weighted PI index of tensor product and strong product of some graphs are obtained. More »» |

2021 |
T. P. Sandhiya, J. Geetha, and Dr. Somasundaram K., “Total colorings of certain classes of lexicographic product graphs”, Discrete Mathematics, Algorithms and Applications, p. 2150129, 2021.[Abstract] A total coloring of a graph G is an assignment of colors to all the elements (vertices and edges) of the graph in such a way that no two adjacent or incident elements receive the same color. The Total Chromatic Number, χ″(G) is the minimum number of colors which need to be assigned to obtain a total coloring of the graph G. The Total Coloring Conjecture made independently by Behzad and Vizing claims that, Δ(G) + 1 ≤ χ″(G) ≤ Δ(G) + 2, where Δ(G) represents the maximum degree of G. The lower bound is sharp, the upper bound remains to be proved. In this paper, we prove the Total Coloring Conjecture for certain classes of lexicographic product and deleted lexicographic product of graphs. More »» |

2020 |
M. S. and Dr. Somasundaram K., “Total coloring of the prismatic graphs”, Discrete Mathematics, Algorithms and Applications, vol. 12, p. 2050032, 2020.[Abstract] A total coloring of a graph is an assignment of colors to all the elements of the graph such that no two adjacent or incident elements receive the same color. A graph G is prismatic, if for every triangle T, every vertex not in T has exactly one neighbor in T. In this paper, we prove the total coloring conjecture (TCC) for prismatic graphs and the tight bound of the TCC for some classes of prismatic graphs. More »» |

2020 |
Thushara M. G. and Dr. Somasundaram K., “Forward and backward static analysis for critical numerical accuracy in floating-point programs”, Computer Science, vol. T. 21 (2), pp. 179–192, 2020.[Abstract] In this article, we introduce a new static analysis for numerical accuracy. We address the problem of determining the minimal accuracy on the inputs and on the intermediary results of a program containing foating-point computations in order to ensure a desired accuracy on the outputs. The main approach is to combine a forward and a backward static analysis, done by abstract interpretation. The backward analysis computes the minimal accuracy needed for the inputs and intermediary results of the program in order to ensure a desired accuracy on the results, specied by the user. In practice, the information collected by our analysis may help to optimize the formats used to represent the values stored in the variables of the program or to select the appropriate sensors. To illustrate our analysis, we have shown a prototype example with experimental results. More »» |

2020 |
R. Vignesh, Mohan, S., J. Geetha, and Dr. Somasundaram K., “Total colorings of core-satellite, cocktail party and modular product graphs”, TWMS Journal of Applied and Engineering Mathematic, vol. 10, no. 3, pp. 778-787, 2020.[Abstract] A total coloring of a graph G is a combination of vertex and edge colorings of G. In other words, is an assignment of colors to the elements of the graph G such that no two adjacent elements (vertices and edges) receive a same color. The total chromatic number of a graph G, denoted by χ00(G), is the minimum number of colors that suffice in a total coloring. Total coloring conjecture (TCC) was proposed independently by Behzad and Vizing that for any graph G, ∆(G) + 1 ≤ χ00(G) ≤ ∆(G) + 2, where ∆(G) is the maximum degree of G. In this paper, we prove TCC for Core Satellite graph, Cocktail Party graph, Modular product of paths and Shrikhande graph. More »» |

2020 |
J. Geetha, Dr. Somasundaram K., and Hung-Lin Fu, “Total colorings of circulant graphs”, Discrete Mathematics, Algorithms and Applications, p. 2150050, 2020.[Abstract] The total chromatic number χ″(G) is the least number of colors needed to color the vertices and edges of a graph G such that no incident or adjacent elements (vertices or edges) receive the same color. Behzad and Vizing proposed a well-known total coloring conjecture (TCC): χ″(G) ≤ Δ(G) + 2, where Δ(G) is the maximum degree of G. For the powers of cycles, Campos and de Mello proposed the following conjecture: Let G = Cnk denote the graphs of powers of cycles of order n and length k with 2 ≤ k < ⌊n 2 ⌋. Then, χ″(G) = Δ(G) + 2, if k > n 3 − 1 and n is odd; and Δ(G) + 1, otherwise. In this paper, we prove the Campos and de Mello’s conjecture for some classes of powers of cycles. Also, we prove the TCC for complement of powers of cycles. More »» |

2020 |
S. Mohan, J. Geetha, and Dr. Somasundaram K., “Total coloring of quasi-line graphs and inflated graphs”, Discrete Mathematics, Algorithms and Applications, vol. 2150060, p. 2150060, 2020.[Abstract] A total coloring of a graph is an assignment of colors to all the elements (vertices and edges) of the graph such that no two adjacent or incident elements receive the same color. A claw-free graph is a graph that does not have K1,3 as an induced subgraph. Quasi-line and inflated graphs are two well-known classes of claw-free graphs. In this paper, we prove that the quasi-line and inflated graphs are totally colorable. In particular, we prove the tight bound of the total chromatic number of some classes of quasi-line graphs and inflated graphs. More »» |

2019 |
Dr. Somasundaram K. and Calicut, C., “A routing algorithm and a router architecture for 3D NoC”, Computer Science, vol. 20, no. 3, 2019.[Abstract] In recent years, the enhancement of microchip technologies has enabled large scale Systems-on-Chip (SoC). Due to sharp increase in number of processing elements, SoC faces various challenges in design and testing. Network on Chip (NoC) is an alternative technology to overcome the challenges in SoC design and testing. NoC emerged as a key architecture that allows one to optimize the parameters like power and area. In spite of its applications, NoC faces some real time challenges like designing an optimum topology, routing scheme and application mappings. In this paper, we address the main three issues on NoC, namely, designing of an optimal topology, routing algorithm and a router design for the topology. First, we propose a topology and a routing algorithm. We prove that our recursive network topology is Hamiltonian connected and we propose an algorithm for data packet transmissions, which is free from cyclic deadlocks and the algorithm maximizes the congestion factor. Our experimental results show that the proposed topology gives better performance in terms of average latency and power than the other topologies. Finally, we propose a router architecture for our 3D-NoC. The router architecture is based on shared buffers. Also, our experimental results indicate that the proposed router architecture consumes less area and power than the Virtual Channel architecture. More »» |

2019 |
Thushara M. G. and Dr. Somasundaram K., “Static Analysis on Floating-Point Programs Dealing with Division Operations”, International Journal of Advanced Computer Science and Applications, vol. 10, 2019. |

2019 |
R. Vignesh, J. Geetha, and Dr. Somasundaram K., “Total coloring conjecture for vertex, edge and neighborhood corona products of graphs”, Discrete Mathematics, Algorithms and Applications, vol. 11, p. 1950014, 2019.[Abstract] A total coloring of a graph G is an assignment of colors to the elements of the graph G such that no adjacent vertices and edges receive the same color. The total chromatic number of a graph G, denoted by χ″(G), is the minimum number of colors that suffice in a total coloring. Behzad and Vizing conjectured that for any simple graph G, Δ(G) + 1 ≤ χ″(G) ≤ Δ(G) + 2, where Δ(G) is the maximum degree of G. In this paper, we prove the tight bound of the total coloring conjecture for the three types of corona products (vertex, edge and neighborhood) of graphs. More »» |

2018 |
Thushara M. G., Dr. Somasundaram K., and Jayaraj Poroor, “Analysis of Numerical Accuracy in Floating Point Programs Using Abstract Interpretation”, AFMSS, Springer LNCS (in print), 2018. |

2018 |
Nallasamy Viswanathan, Kuppusamy Paramasivam, and Dr. Somasundaram K., “Vertical Links Minimized 3D NoC Topology and Router-Arbiter Design”, INTERNATIONAL ARAB JOURNAL OF INFORMATION TECHNOLOGY, vol. 15, pp. 469–478, 2018.[Abstract] Design of a topology and its router plays a vital role in a 3D Network-on-Chip (3D NoC) architecture. In this paper, we develop a partially vertically connected topology, so called 3D Recursive Network Topology (3D RNT) and using an analytical model, we study the performance of the 3D RNT. Delay per Buffer Size (DBS) and Chip Area per Buffer Size (CABS) are the parameters considered for the performance evaluation. Our experimental results show that the vertical links are cut down upto 75% in 3D RNT compared to that of 3D Fully connected Mesh Topology (3D FMT) at the cost of increasing DBS by 8%, besides 10% lesser CABS is observed in the 3D RNT. Further, a Programmable Prefix router-Arbiter (PPA) is designed for 3D NoC and its performance is analyzed. The results of the experimental analysis indicate that PPA has lesser delay and area (gate count) compared to Round Robin Arbiter (RRA) with prefix network More »» |

2018 |
R Vignesh, J. Geetha, and Dr. Somasundaram K., “Total Coloring Conjecture for Certain Classes of Graphs”, Algorithms, vol. 11, p. 161, 2018.[Abstract] A total coloring of a graph G is an assignment of colors to the elements of the graph G such that no two adjacent or incident elements receive the same color. The total chromatic number of a graph G, denoted by , is the minimum number of colors that suffice in a total coloring. Behzad and Vizing conjectured that for any graph G, , where is the maximum degree of G. In this paper, we prove the total coloring conjecture for certain classes of graphs of deleted lexicographic product, line graph and double graph. More »» |

2018 |
J. Geetha and Dr. Somasundaram K., “Total Colorings of Product Graphs”, Graphs and Combinatorics (2018), vol. 34, no. 2, pp. 339–347, 2018.[Abstract] A total coloring of a graph is an assignment of colors to all the elements of the graph in such a way that no two adjacent or incident elements receive the same color. In this paper, we prove Behzad–Vizing conjecture for product graphs. In particular, we obtain the tight bound for certain classes of graphs. More »» |

2017 |
S. Mohan, J. Geetha, and Dr. Somasundaram K., “Total coloring of the corona product of two graphs”, AUSTRALASIAN JOURNAL OF COMBINATORICS, vol. 68, no. 1, pp. 15–22, 2017.[Abstract] A total coloring of a graph is an assignment of colors to all the elements (vertices and edges) of the graph such that no two adjacent or incident elements receive the same color. In this paper, we prove the tight bound of the Behzad and Vizing conjecture on total coloring for the corona product of two graphs G and H, when H is a cycle, a complete graph or a bipartite graph. More »» |

2016 |
S. Mohan, J. Geetha, and Dr. Somasundaram K., “Total Coloring of Certain Classes of Product Graphs”, Electronic Notes in Discrete Mathematics, vol. 53, pp. 173 - 180, 2016.[Abstract] Abstract A total coloring of a graph is an assignment of colors to all the elements of the graph such that no two adjacent or incident elements receive the same color. In this paper, we prove the tight bound of Behzad and Vizing conjecture on total coloring for Compound graph of G and H, where G and H are any graphs. More »» |

2016 |
J. Geetha and Dr. Somasundaram K., “Total Chromatic Number and Some Topological Indices”, Electronic Notes in Discrete Mathematics, vol. 53, pp. 363 - 371, 2016.[Abstract] Abstract The total chromatic number χ ″ ( G ) of G is the smallest number of colors needed to color all elements of G in such a way that no adjacent or incident elements get the same color. The harmonic index H ( G ) of a graph G is defined as the sum of the weights 2 d ( u ) + d ( v ) of all edges uv of G, where d ( u ) denotes the degree of the vertex u in G. In this paper, we show a relation between the total chromatic number and the harmonic index. Also, we give relations between total chromatic number and some topological indices of a graph. More »» |

2016 |
P. Subramanian and Dr. Somasundaram K., “Some Conjectures on Permanents of Doubly Stochastic Matrices”, Journal of Discrete Mathematical Sciences and Cryptography, vol. 19, pp. 997-1011, 2016.[Abstract] AbstractLet denote the set of all doubly stochastic matrices of order n. Foregger [3] raised a n question whether per per (A) holds for all and , where Jn is the n × n matrix with each entry equal to .But this inequality does not hold good for all matrices in general. In this paper, we consider the above inequality for subpermanents and we provide a sufficient condition for a matrix A ∈ Ωn to satisfy the inequality σk(fJn + (1−t)A) ≤ σk(A) for 0 ≤ t ≤ 1 and discuss the consequences of this inequality. More »» |

2016 |
Ka Chythanya and Dr. Somasundaram K., “Evaluation of partially connected mesh topology and shared queues router architecture for network on chip”, International Journal of Control Theory and Applications, vol. 9, pp. 4393-4399, 2016.[Abstract] Due to unremitting scaling down of CMOS technology, huge number of mixed devices is integrated on a chip. For the efficient communication between these devices routers are employed. Buffers are placed on either the input or in the output ports of a Network on Chip router. If a contention occurs in the output physical channel then input port holds the packet temporarily. During a traffic trace only 40% of the buffers are active and rest are left inactive. Power budgets and area requirements of buffers are high. This leads to the motivation for designing a router architecture with shared queues (RoSHAQ). RoSHAQ improves the buffer utilization by sharing of multiple buffers between different ports. In this paper, we projects a 3 dimensional Partial Mesh of Grid topology (PMG)and we use RoSHAQfor our design. We synthesises our topology using Network Simulator-2 tool. Experimental result concludes that PMG has improved performance in terms of latency, throughput and area in comparison with other topologies. © International Science Press. More »» |

2016 |
P. Mohan, Dr. Somasundaram K., Dash, S. S., Das, S., and Bhaskar, M. A., “Design and evaluation of 3D NoC routers with quality-of-service (QoS) mechanism for multi-core systems”, Advances in Intelligent Systems and Computing, vol. 394, pp. 429-441, 2016.[Abstract] The importance of on-chip communication interconnects was greatly highlighted with the advent of semiconductor technology at nanoscale domain. As the sizes of semiconductor features are reduced day by day, there occurred problems related to wiring. Network-on-Chip architectures are therefore implemented to overcome the wiring issues and have lately been considered as an important area for research. The communication on NoC is carried out in specific topologies by means of routers. In this paper, Partial Mesh of Grid topology (PMG) is considered. We use the Quality of Service (QoS) mechanism to minimize the area and power. PMG-based NoC will give minimum area and power and it reduces the high chances of redundant connections. Throughput and latency are analysed along with other parameters like packet loss ratio and jitter using network simulator NS-2. Area and power analyses are done using Synopsys Design Compiler and PrimeTime PX tool. Our experimental results show that the architecture with QoS mechanism gives a significant reduction in area and power when compared to Region-based Routing (RBR) mechanism. Moreover, the partial mesh of grid topology gives minimal latency and high throughput when compared to mesh of grid topology. © Springer India 2016. More »» |

2015 |
J. Geetha and Dr. Somasundaram K., “Total coloring of generalized sierpiński graphs”, Australasian Journal of Combinatorics, vol. 63, no. 1, pp. 58-69, 2015.[Abstract] A total coloring of a graph is an assignment of colors to all the elements of the graph in such a way that no two adjacent or incident elements receive the same color. In this paper, we prove the tight bound of the Behzad and Vizing conjecture on total coloring for the generalized Sierpiński graphs of cycle graphs and hypercube graphs. We give a total coloring for the WK-recursive topology, which also gives the tight bound. © 2015,University of Queensland. All rights reserved. More »» |

2015 |
Aa Gupta, J. Geetha, and Dr. Somasundaram K., “Total coloring algorithm for graphs”, Applied Mathematical Sciences, vol. 9, pp. 1297-1302, 2015.[Abstract] A total coloring of a graph is an assignment of colors to all the elements of graph such that no two adjacent or incident elements receive the same color. Behzad and Vizing conjectured that for any graph G the following inequality holds: Δ(G)+1≤X′′(G)≤Δ(G)+2, where Δ(G) is the maximum degree of G. Total coloring algorithm is a NP-hard algorithm. In this paper, we give the total coloring algorithm for any graph and we discuss the complexity of the algorithm. © 2015 Aman Gupta, J. Geetha and K. Somasundaram. More »» |

2014 |
Dr. Somasundaram K., Plosila, Jb, and Viswanathan, Nc, “Deadlock free routing algorithm for minimizing congestion in a Hamiltonian connected recursive 3D-NoCs”, Microelectronics Journal, vol. 45, pp. 989-1000, 2014.[Abstract] Network on Chip (NoC) has been proposed as a solution for addressing the challenges in System on Chip (SoC) design. Designing a topology and its routing schemes are vital problems in a NoC. One of the major challenges that designers face today in 3D integration is how to route the data packets within a layer and across the layers in a scalable and efficient manner. In any 3D topology, minimizing the amount of data packet transmissions during the routing is still an open problem. Any efficient traditional routing schemes should avoid deadlocks and minimize network congestion from a source node to a destination node. In this paper, we propose a 3D recursive hyper graph Hamiltonian connected network and we propose a deadlock free routing algorithm to minimize congestion in the network. We show that the proposed topology outperforms the topology presented by Dubois et al. (Elevator-First: a Dealock-free distributed routing algorithm for vertically partially connected 3d-Nocs, IEEE Trans. Comput. 62(3) (2013) 609-615) [1] with respect to average network latency. Also, we analysis the delay bound of the switches for the proposed topology and 3D Partially connected Mesh Topology (PMT) and conclude that our topology performs better than 3D PMT. © 2014 Elsevier Ltd. More »» |

2014 |
Na Viswanathan, Paramasivam, Kb, and Dr. Somasundaram K., “An optimised 3D topology for on-chip communications”, International Journal of Parallel, Emergent and Distributed Systems, vol. 29, no. 4, pp. 346-362, 2014.[Abstract] Increasing system complexity, energy and device reliability, requirement of modular approach, structured layout, effective spatial reuse of resources, scalability and re-programmability have made network-on-chip (NoC) an obvious interconnection design alternative to the ubiquitous bus based on chip communication architecture in system-on-chip. Designing of a topology and its routing scheme plays a vital role in determining performance of any NoC architecture. In recent years, 3D stacked NoC architecture attracts added interest in NoC design as it offers improved performance and shorter global interconnect. In this paper, we have developed a partially, vertically interconnected 3D topology, namely 3D Recursive Network Topology (3D RNT) and prove that the topology has a Hamiltonian connectedness. We have developed deadlock-free routing algorithm for the 3D RNT topology. Also, we compare the performance of the 3D RNT with partially and fully connected 3D mesh topologies (3D PMT and 3D FMT) by conducting suitable experiments. The experiment results show that there is not much deviation in respect of the performance of the 3D RNT on comparing with 3D PMT and 3D FMT even though a number of vertical links are trimmed down to 75%, which is an encouraging outcome as far as design space is concerned. © 2013 © 2013 Taylor & Francis. More »» |

2013 |
P. Subramanian and Dr. Somasundaram K., “Function Mates”, Journal of Algebra and Applied Mathematics, vol. 11, pp. 29-35, 2013.[Abstract] Let Ωκn denote the set of all generalized doubly stochastic matrices with row and column sums are k. Let f be a function from Ω κn ro R, two unequal matrices A and B in Ω κn are called function mates if the function is constant on the line segment tA+(1-t)B, 0 ≤ t ≤ 1, connecting A and B. If two matrices A and B are in affine space, then we derived the conditions for A and B are function mates with specific functions permanent and determinent. More »» |

2013 |
N. Viswanathan, Paramasivam, K., and Dr. Somasundaram K., “Performance and cost metrics analysis of a 3D NoC topology using network calculus”, Applied mathematical sciences, vol. 7, pp. 4173-4184, 2013.[Abstract] The packet switching based Network-on-Chip (NoC) is an obvious interconnect design alternative to the shared bus, crossbar or ring based on-chip communication architecture used in System-on-Chips (SoCs). The advent of the three dimensional NoC (3D NoC) architecture attracts added interest as it offers improved performance and shorter global interconnect. In the 3D NoC architecture, topology plays a vital role in determining the performance of the interconnect architecture. The performance and cost metrics of the 3D NoC topology are evaluated by using simulation in general. However, analytical models provide more insights on how the traffic related parameters influence the performance of the topology. In this paper, the traffic related parameters of a 3D NoC topology, namely 3D Recursive Network Topology (3D RNT) are evaluated by using network calculus based methodology and the results of the evaluation are compared against the results produced using simulation. More »» |

2013 |
Na Viswanathan, Paramasivam, Kb, and Dr. Somasundaram K., “Performance comparison of 3D NoC topologies using network calculus”, Life Science Journal, vol. 10, pp. 4379-4385, 2013.[Abstract] Nowadays, System-on-Chips (SoCs) designers are forced to integrate tens to hundreds of functional and storage blocks in a single die to implement emerging complex computation, multimedia and network services. The integration of huge degree of the blocks in a single die poses new challenge in designing the interconnect architecture of the blocks in SoCs. The traditional bus based interconnect infrastructure is ineffective as the number of the blocks increases more than ten. The packet based Network- on- Chip (NoC) is an obvious interconnect design alternative to the bus based on-chip communication architecture in SoCs. The advent of 3D NoC architecture attracts added interest as it offers improved performance and shorter global interconnect. Evolving an efficient 3D network topology and developing 3D routing scheme play a crucial role in determining the performance of 3D NoC interconnect architecture. In this paper, two 3D NoC topologies, namely 3D Recursive Network Topology (3D RNT) and 3D Modified Mesh Topology (3D MMT) are presented. End-to-end delay, switch buffer size and the influence of the buffer size in determining area overhead requirement of the two topologies are evaluated using an analytical model of network calculus and the evaluation results of the two topologies are compared. It is shown that the 3D RNT outperforms the 3D MMT even though 50% of the vertical links are trimmed down in the former topology. Further, 20 % reduction in average switch buffer size and 16% reduction in area overhead requirement are achieved in the 3D RNT. The results of the analysis are of use to evaluate and optimize 3D NoC interconnect architecture as far as the design space is concerned. More »» |

2012 |
Dr. Somasundaram K. and Plosila, Jc, “Deadlock free routing algorithm for minimizing data packet transmission in network on chip”, International Journal of Embedded and Real-Time Communication Systems, vol. 3, pp. 70-81, 2012.[Abstract] Network on chip (NoC) has been proposed as a solution for addressing the design challenges of future high performance nanoscale architectures. In NoCs, the traditional routing schemes are routing packets through a single path or multiple paths from one source node to a destination node, minimizing the congestion in the routing architecture. Although these routing algorithms are moderately efficient, they are time dependent. To reduce overall data packet transmission time in the network, the authors consider a network with multiple sources and multiple destinations. Multi-dimensional routing problems appear naturally in several resource allocation problems, communication networks and wireless sensor networks. In this paper, the authors have constructed a deadlock-free multi-dimensional path routing algorithm for minimizing the congestion in NoC. More »» |

2012 |
Na Viswanathan, Paramasivam, Kb, and Dr. Somasundaram K., “Exploring optimal topology and routing algorithm for 3D network on chip”, American Journal of Applied Sciences, vol. 9, pp. 300-308, 2012.[Abstract] Problem statement: Network on Chip (NoC) is an appropriate candidate to implement interconnections in SoCs. Increase in number of IP blocks in 2D NoC will lead to increase in chip area, global interconnect, length of the communication channel, number of hops transversed by a packet, latency and difficulty in clock distribution. 3D NoC is evolved to overcome the drawbacks of 2D NoC. Topology, switching mechanism and routing algorithm are major area of 3D NoC research. In this study, three topologies (3D-MT, 3D-ST and 3D-RNT) and routing algorithm for 3D NoC are presented. Approach: Experiment is conducted to evaluate the performance of the topologies and routing algorithm. Evaluation parameters are latency, probability and network diameter and energy dissipation. Results: It is demonstrated by a comparison of experimental results analysis that 3D-RNT is a suitable candidate for 3D NoC topology. Conclusion: The performance of the topologies and routing algorithm for 3D NoC is analysed. 3D-MT is not a suitable candidate for 3D NoC, 3D-ST is a suitable candidate provided interlayer communications are frequent and 3D-RNT is a suitable candidate as interlayer communications are limited. © 2012 Science Publications. More »» |

2011 |
Dr. Somasundaram K. and Subramanian, P., “Determinantal Mates”, International Journal of Algebra, vol. 5, pp. 1181–1187, 2011. |

2010 |
V. H. Prathyush and Dr. Somasundaram K., “Multi-level sequential circuit partitioning for test vector generation for low power test in VLSI”, International Journal of Engineering, Science and Technology, vol. 2, 2010.[Abstract] Sequential graph partitioning algorithms have been developed to fulfill the requirements of emerging multi-phase problems in circuit testing models. In this paper, we present a multi-level graph partitioning algorithm for circuit partitioning, which will minimize the number of test vectors during a low power test in VLSI circuits. By reducing the number of test vectors, we can reduce the energy consumption during the test. Our experimental results with ISCAS bench mark circuits have shown that the power can be reduced up to 55%. More »» |

2009 |
Dr. Somasundaram K., “Graph Mates”, Electronic Notes in Discrete Mathematics, vol. 33, pp. 37-41, 2009.[Abstract] A weighted digraph graph D is said to be doubly stochastic if all the weights of the edges in D are in [0, 1] and sum of the weights of the edges incident to each vertex in D is one. Let Ω(G) be denoted as set of all doubly stochastic digraphs with n vertices. We defined a Graph Mates in Ω(G) and derived a necessary and sufficient condition for two doubly stochastic digraphs are to be a Graph Mates. © 2009 Elsevier B.V. All rights reserved. More »» |

2008 |
Dr. Somasundaram K., “Permanent: Evolution by Parallel Algorithm”, Journal of Information and Computing Science, vol. 3, no. 1, pp. 69-72, 2008. |

2008 |
Dr. Somasundaram K. and Raj, S. M. Ab, “Permanent: Evaluation by parallel algorithm”, Journal of Information and Computing Science, vol. 3, no. 1, pp. 69-72, 2008.[Abstract] Permanent of a matrix is # p - complete problem shown by many authors. In this paper we present a parallel algorithm for evaluation of permanent of an n × n matrix with multi processors. More »» |

2008 |
SaMaria Arulraj and Dr. Somasundaram K., “Permanental Mates: Perturbations and Hwang's conjecture”, Applied Mathematics Letters, vol. 21, pp. 786-790, 2008.[Abstract] Let Ωn denote the set of all n × n doubly stochastic matrices. Two unequal matrices A and B in Ωn are called permanental mates if the permanent function is constant on the line segment t A + (1 - t) B, 0 ≤ t ≤ 1, connecting A and B. We study the perturbation matrix A + E of a symmetric matrix A in Ωn as a permanental mate of A. Also we show an example to disprove Hwang's conjecture, which states that, for n ≥ 4, any matrix in the interior of Ωn has no permanental mate. © 2007 Elsevier Ltd. All rights reserved. More »» |

2007 |
S. Maria Arulraj and Dr. Somasundaram K., “Star Matrices: Properties And Conjectures”, Applied Mathematics E-Notes, vol. 7, pp. 42–49, 2007.[Abstract] Let Ωn denote the set of all nxn doubly stochastic matrices. A matrix B ∈ Ωn is said to be a star matrix if per(αB + (1 − α)A) ≤ α perB + (1 − α)perA, for all A ∈ Ωn and for all α ∈ [0, 1]. In this paper we derive a necessary condition for a star matrix to be in Ωn, and a partial proof of the star conjecture: The direct sum of two star matrices is a star matrix. More »» |

2007 |
Dr. Somasundaram K., “Multi-level Sequential circuit partitioning for delay minimization of VLSI circuits”, J. of Information and Computing Science, vol. 2, pp. 66–70, 2007.[Abstract] Sequential graph partitioning algorithms have been developed to fulfill the requirements of emerging multi-phase problems in circuit delay models. In this paper we propose a heuristic algorithm for kpartition, which minimizes the circuit delay and cut size. Experimental results with MCNC benchmark circuits have shown that the delay in the circuit can be reduced by marginally in comparison with the other algorithms More »» |

2006 |
Dr. Somasundaram K., “Graph Partitioning in VLSI Design and Testing: A Review”, Discrete Mathematics and Its Applications, p. 162, 2006. |

2005 |
Bapat and Dr. Somasundaram K., “Cremer’s Rule for Generalized Matrix Functions”, Bulletin of Kerala Mathematics Association, vol. 2, no. 2, pp. 1-7, 2005. |

2002 |
M. Arul S. Raj and Dr. Somasundaram K., “Results on Permanents and Graphs”, Mathematics Preprint Archive, no. 10, 2002.[Abstract] In this paper the sufficient condition for a graph to be Hamiltonian and non-Hamiltonian are discussed. Some simple results on tree with permanent of A(T) and X(T). If G is a Bipartite graph with #X = #Y = n such that per A(G) = (n!), then G is a complete bipartite graph. Also some results in permanent of A(G) of separable graphs, equivalence and partial order relations are discussed. More »» |

1994 |
RC Goyal, AT Kulkarni, and Dr. Somasundaram K., “AIDS Awareness among rural Community-a study”, Swasth Hind, vol. 18, pp. 262–266, 1994. |

1994 |
RC Goyal, NL Sachdeva, and Dr. Somasundaram K., “Oral health status of rural community in Western Maharashtra”, Ind J PSM, vol. 25, pp. 137–45, 1994. |

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2019 |
M. T. Indu, Dr. Somasundaram K., and Dr. Shunmuga Velayutham C., “A Preliminary Investigation on a Graph Model of Differential Evolution Algorithm”, in Pattern Recognition and Machine Intelligence, Cham, 2019.[Abstract] <p>Graph based analysis of Evolutionary Algorithms (EAs), though having received little attention, is a propitious method of analysis for the understanding of EA behavior. However, apart from mere proposals in literature, the graph model(s) of EAs have not been aptly demonstrated of their full potential. This paper presents a critical analysis of an existing prominent graph model of EAs. The comprehensive analysis involved modelling as graphs, the interactions within Differential Evolution (DE) algorithms (DE/rand/1/bin and DE/best/1/bin by way of example), while solving CEC2005 benchmark functions. Subsequently, the graph-theoretic measures- degree centrality, clustering coefficient, betweenness centrality and closeness centrality- were deployed to investigate the potential of graph model in providing insights about EA dynamics. However, the model falls short in providing useful insights about the DE runs. The observed insights were qualitative in nature and not competitive compared to empirical analysis. This indicates the need for considerable research attention in designing robust graph models.</p> More »» |

2019 |
Dr. Somasundaram K., “Design of a Virtual Channel Router Architecture for Low Power on Mesh-of-Grid Topology for Network on Chip”, in Proceeding of International Conference on Applied Soft Computing and Communication Networks (ACN 2019), LNNS 125, 63-79, 2019.[Abstract] System on chip (SoC) cannot handle the large communication systems due to increasing number of applications. Network on chip (NoC) is an alternative technology to system on chip (SoC) to improve the performance. Still, NoC faces a lot of challenges in its design and testing. The main challenges are power, area, and latency constraints. Router places an important role in communication between the cores. Design of a router is an integral part of a NoC system design. In this paper, we have considered the mesh-of-grid (MoG) topology. We have proposed an architecture for virtual channel router (VCR) to reduce the area and power. Region-based routing (RBR) mechanism is used here to optimize the area and power. We have evaluated various network parameters using the Network Simulator-2. Our experimental results show that the architecture with RBR mechanism gives a significant reduction in area and power in comparison with table-based mechanism and other mechanism. We have shown that the MoG topology gives better latency and packet delivery ratio than the mesh topology. More »» |

2015 |
J. Geetha and Dr. Somasundaram K., “Total Coloring for Certain Classes of Claw-free Graphs”, in International Conference on Graph Theory and its Applications, Puducherry, 2015. |

2011 |
V. N., K., P., and Dr. Somasundaram K., “Performance analysis of cluster based 3D routing algorithms for NoC”, in 2011 IEEE Recent Advances in Intelligent Computational Systems, 2011, pp. 157-162.[Abstract] In the nano scaled transistors integration era, interconnection of IP blocks and data exchange among the IP blocks are crucial concerns in System on Chip (SoC). Network-on-Chip (NoC) is an on-chip communication methodology proposed to resolve the increased interconnection problems in SoC. In deep sub-micron regime, 3D NoC becomes an emerging research area in recent years as the three dimensional (3D) integrated circuits (ICs) can offer shorter interconnection wire and dissipate lesser power. Major area of the 3D NoC research is network topology and routing techniques. In this paper, we present an NS-2 (Network Simulator) simulation environment for two 3D network topologies (GBT and CBT) and cluster based routing algorithms. Simulation results are reported. Simulation results about the relationship between switch buffer size, injected traffic load, packet delay, packet drop probability and energy dissipation are analyzed. On comparing CBT with GBT, a significant performance improvement is demonstrated. More »» |

2009 |
Dr. Somasundaram K. and Plosila, J., “Multi-dimensional routing algorithms for congestion minimization in Network-on-Chip”, in 2009 NORCHIP, 2009.[Abstract] In network on chip (NoC), the traditional routing schemes are routing the network through a single path or multiple paths from one source node to a destination node, which will minimize the congestion in the routing architecture. Though these routing algorithms are moderately efficient, but time dependent. To reduce overall data packet transmission time in the network, we consider a network with multiple sources and multiple destinations. Multi-dimensional routing problems appear naturally in several resource allocation problems, communication networks and wireless sensor networks. In this paper, we have shown a multi-dimensional path routing algorithm for minimizing the congestion in NoC with deadlock free. More »» |

2004 |
Dr. Somasundaram K., “Graph Partitioning Algorithms in VLSI Design and Testing: Review”, in Proceeding of Discrete Mathematics and its Applications, 162-168, 2004. |

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2014 |
N. V. Anjali and Dr. Somasundaram K., “Design and evaluation of virtual channel router for mesh-of-grid based NoC”, International Conference on Electronics and Communication Systems (ICECS), 2014 . Coimbatore, 2014.[Abstract] Network on Chips (NoCs) has now replaced the bus based architectures for communication between different cores in a multiprocessor System on Chip (SoC). NoC integrates SoCs in a better manner. It has the advantage of good scalability and high bandwidth. The communication on NoC is carried out by means of routers. Routers are the back bone of NoC. The design of routers is different for different topologies. In this paper, a Mesh-of-Grid topology is considered. A virtual channel router for mesh of grid topology of NoC is presented here. Area and Power is synthesized for the virtual channel router using Synopsys Design Vision. The experimental results show that the area and power will increase if the bit size of flit is increased. More »» |

2013 |
Dr. Somasundaram K. and Subramainan, P., “A note on some conjectures in permanent”, 3rd-India-Taiwan Conference on Discrete Mathematics, NCTU, November 18-22. 2013. |

2012 |
Na Viswanathan, Paramasivam, Kb, and Dr. Somasundaram K., “Exploring hierarchical, cluster based 3D topologies for 3D NoC”, Procedia Engineering, vol. 30. Coimbatore, pp. 606-615, 2012.[Abstract] Network-on-Chip (NoC) has been recognized as an effective solution for complex on-chip communication problems faced in System-on-Chips (SoCs). Network topology, switching mechanism and routing algorithms are the key research area in NoC. In recent years, since the inception of Through-Silicon-Vias (TSVs) to realize vertical channel, 3D stacked NoC architecture attracts a lot of interest as it offers improved performance and shorter global interconnect. In this paper, two clustered 3D network topologies (3D-ST and 3D-RNT) and hierarchical, cluster based routing algorithms are presented. Experimental results on various parameters like latency, drop probability and energy dissipation are compared for the two topologies. It is demonstrated from the analysis that 3D-RNT is an appropriate candidate for 3D NoC provided interlayer communications are not very frequent. More »» |

2003 |
M. Arulraj and Dr. Somasundaram K., “Results on Permanents and Graphs”, International Conference on Graph theory, Simon Fraser University, Vancouver, Canada. 2003. |

2000 |
M. Arulraj and Dr. Somasundaram K., “Permanents and Graphs”, Proceeding of UGC and CSIR Conference on Applications of Mathematics. 2000. |

1998 |
M. Arulraj and Dr. Somasundaram K., “Permanents and Graphs”, Proceeding of UGC Conference on Discrete Mathematics. 1998. |

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