On the Degree-Based Topological Indices of the Tickysim SpiNNaker Model

: Tickysim is a clock tick-based simulator for the inter-chip interconnection network of the SpiNNaker architecture. Network devices such as arbiters, routers, and packet generators store, read, and write forward data through ﬁxed-length FIFO buffers. At each clock tick, every component executes a “read” phase followed by a “write” phase. The structures of any ﬁnite graph which represents numerical quantities are known as topological indices. In this paper, we compute degree-based topological indices of the Tickysim SpiNNaker Model ( TSM ) sheet.


Introduction
Neural networks are applicable in many solutions for classification, prediction, control, etc.The variety of purposes is growing but with each new application the expectations are higher.We want neural networks to be more precise independently of the input data.Efficiency of the processing in a large manner depends on the training algorithm.Basically this procedure is based on the random selection of weights in which neurons connections are burdened.During training process we implement a method which involves modification of the weights to minimize the response error of the entire structure see details in [1,2].
Convolutional neural network (CNN) is an essential model to achieve high accuracy in various machine learning applications, such as image recognition and natural language processing.One of the important issues for CNN acceleration with high energy efficiency and processing performance is efficient data reuse by exploiting the inherent data locality.Recurrent neural networks (RNNs) are powerful models of sequential data.They have been successfully used in domains such as text and speech.However, RNNs are susceptible to overfitting; regularization is important [3].Motivated by these networks we consider the Tickysim SpiNNaker Model(network) for utilization its topological properties.
In this paper all graphs are finite, simple, and undirected.Let V(G) and E(G) be the vertex set and edge set of a graph G.The vertices u, v ∈ V(G) are adjacent (or neighbors) if u and v are endpoints of e ∈ E(G) and e is incident with the vertices u and v and e is said to connect u and v.The set of all neighbors of a vertex u of G denoted by N(u) is called the neighborhood of v.The degree of a vertex in an undirected simple graph is the number of edges incident with it.The degree of the vertex u is denoted by ζ(u) and S u is the sum of degrees of all vertices adjacent to the vertex u.In other words, All the concepts of graph theory and combinatorics are used from the book of Harris et al. [4,5].
The application of molecular structure descriptors is now a standard procedure in the study of structure-property relations, especially in QSPR/QSAR study.In the past couple of years, the amount of proposed nuclear descriptors is rapidly increases as a result of the significance of the creation of these descriptors.They interface the particular physico-substance properties of mixture blends.A most seasoned, considered, and prominent topological record among all degree-based topological lists is the Randić index, which was presented by Randić in 1975 [6].This record was discovered to be reasonable with the end goal of a medication plan [7].The numerical elements of the Randić index incorporates its association with the standardized Laplacian framework [8][9][10].The formal definition of the Randić index of a graph G is given as follows: . ( Soon after the discovery of Randić index, the general Randić index was introduced.It is denoted by R α (G), and its formula is given as: where α is a nonzero real number.Zhou et al. [11] introduced the general sum-connectivity index χ α (G) and defined it as: where α is a real number.Shirdel et al. introduced a new degree-based Zagreb index named the "hyper-Zagreb index" which is defined in [12], and is also known as general sum-connectivity index χ 2 (G).The first general Zagreb index was studied in [13].
Estrada et al. invented the atom-bond connectivity index, abbreviated as the ABC index [14].ABC index is of much importance due to its correlation with the thermodynamic properties of alkanes (see [15,16]).The definition of the ABC index is as follows: ( The fourth version of the ABC index was introduced by Ghorbani and Hosseinzadeh [17], and is defined as: Another important degree-based topological index is the geometric-arithmetic index (GA index) and is of much importance due to its application to acyclic, unicyclic, and bicyclic molecular graphs [18].The formal definition of the GA index is as follows: Recently, the fifth version of GA was introduced by Graovac el al. [19], defined as: In [20], Ghorbani and Azimi defined first multiple Zagreb index PM 1 (G) and second multiple Zagreb index PM 2 (G), defined as: These multiple Zagreb indices are studied for some chemical structures in [21][22][23][24][25].The first Zagreb polynomial M 1 (G, x) and second Zagreb polynomial M 2 (G, x) are defined as:

Applications of Topological Indices
To relate with certain physico-concoction properties, the GA index has much preferred prescient control over the prescient energy of the Randić connectivity index [26].The first and second Zagreb indexes were found to be helpful for calculation of the aggregate π-electron energy of the particles inside particular rough articulations [27].These are among the graph invariants that were proposed for estimation of the skeleton of stretching of the carbon atom [28].The Randić index is a topological descriptor that has related with a great deal of the synthetic qualities of atoms and has been discovered parallel to processing the boiling point and Kovats constants of the particles.The particle bond network (ABC) index gives a decent connection to the security of direct alkanes and also the stretched alkanes and for processing the strain vitality of cyclo alkanes [29,30].
In the past two decades, analysts contemplated certain substance diagrams and arrangement s and processed their particular indices.W. Gao and M. R. Farahani figured degree-based indices of synthetic structures by utilizing an edge-separated technique [31].Gao et al. [32] contemplated concoction structures in medications and some medication structures, and processed the overlooked topological indices.Some different utilizations of the atomic descriptors of sub-atomic diagrams and systems are given in the reference list and the references [33].These applications and writing survey inspired us to investigate some new substance diagrams and process their topological indices [34][35][36][37].

Materials and Methods
At the highest level of abstraction, inter-chip architectures are basically mathematical graphs where each device is considered as vertex and the topology used between these devices reflects the edges and in total nature of graph.One of the network topologies used in this model is 12 × 12 hexagonal torus.In this topology, each node is connected to six incident nodes.We also consider the finite Tickysim SpiNNaker Model sheet which is obtained by hexagonal torus.For more details, see [38].
The Second of the network topology consists of a set of a hexagonal segments of a hexagonal mesh of nodes.Each node in the simulation represents a SpiNNaker chip that contains a router, packet generator, packet consumer, and a tree of two-input round-robin arbiters which arbitrates between the inputs to the router.The router always consists of a four-stage pipeline.If a packet cannot be forwarded to its requested output after 50 cycles at the head of the router, it is dropped.The packet generator generates packets for each node of the system.If the output buffer is full, the packet generator waits until a space becomes available.The packet consumer receives incoming packets immediately, but the packet consumer will wait 10 cycles before accepting another packet.The arbiter tree is based on SpiNNaker's NoC aspects.In each cycle, the arbiter selects a waiting packet on one of its inputs and forwards it to its output if there is space in the output buffer.
The graph TSM of a Tickysim SpiNNaker Model sheet is shown in Figure 1.The number of vertices in the Tickysim SpiNNaker Model sheet are mn, and the vertex partition of the graph TSM sheet based on the degree of vertices is shown in Table 1.

Main Results
Theorem 1.Let TSM be a Tickysim SpiNNaker Model sheet, then where α is a real number.
Proof.The number of edges of the TSM sheet graph is 3mn − 2m − 2m + 1.The edge partition based on the degree of the end vertices of each edge are shown in Table 2. Since the formula of the general Randić index is and the formula of the general sum-connectivity index is This completes the proof. Proof.
This completes the proof.
In the next two theorems, we calculated the fourth atom-bond connectivity index ABC 4 and the fifth geometric-arithmetic index GA 5 .There are eighteen types of edges on the degree-based sum of neighbors vertices of each edge in the Tickysim SpiNNaker Model sheet.We used this partition of edges to calculate ABC 4 and GA 5 indices.Table 3 gives such types of edges of the Tickysim SpiNNaker Model sheet.The edge set E(TSM) is divided into eighteen edge partitions based on the degree of end vertices.The edge partition E u,v (TSM) contains m u,v edges uv, where S u = u, S v = v, and m u,v = |E u,v (TSM)|.3. The fifth geometric-arithmetic index GA 5 is defined as: This implies that This completes the proof.
We compute the hyper-Zagreb index HM(G), first multiple Zagreb index PM 1 (G), second multiple Zagreb index PM 2 (G), and Zagreb polynomials M 1 (G, x), M 2 (G, x) for the Tickysim SpiNNaker Model sheet in the following theorem.
By inserting the values, we obtain This completes the proof.

Conclusions
In this paper, we deal with a Tickysim SpiNNaker Model sheet and study its topological indices.We determined the first general Zagreb index M α , general Randić connectivity index R α , general sum-connectivity index χ α , atom-bond connectivity index ABC, geometric-arithmetic index GA, fourth atom-bond connectivity index ABC 4 , fifth geometric-arithmetic index GA 5 , hyper-Zagreb index HM(G), first multiple Zagreb index PM 1 (G), second multiple Zagreb index PM 2 (G), and Zagreb polynomials M 1 (G, x), M 1 (G, x).
In the future, we are interested in designing some incipient architectures/networks and then studying their topological indices, which will be quite auxiliary to understanding their underlying topology.

Figure 1 .
Figure 1.Graph of the Tickysim SpiNNaker Model sheet for m = n = 12.

Table 1 .
The vertex partition of the graph TSM sheet based on the degree of vertices.

Table 2 .
The edge partition of a graph TSM sheet based on the degree of end vertices of each edge.