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The quadratic embedding (QE) class of the connected graph G = (V, E) is determined based on its quadratic embedding constant (QEC) of the distance matrix graph G. The distance matrix is a conditionally definite negative, or equivalently if it admits a quadratic embedding in a Hilbert space, or if QEC of the graph G is non-positive. In this study, the classification for the hairy cycle graphs such as bearded cycle graph BC(k,m) for k is even, bearded cycle graph BC(3,m) and broken sun graph BC(4,m) will be calculated, and we derive the formulae of its QEC. These classes of graphs belongs to the QE class.

A modular irregular graph is a graph that admits a modular irregular labeling. A modular irregular labeling of a graph G of order n is a mapping of the set of edges of the graph to 1,2,…,k such that the weights of all vertices are different. The vertex weight is the sum of its incident edge labels, and all vertex weights are calculated with the sum modulo n. The modular irregularity strength is the minimum largest edge label such that a modular irregular labeling can be done. In this paper, we construct a modular irregular labeling of two classes of graphs that are biregular; in this case, the regular double-star graph and friendship graph classes are chosen. Since the modular irregularity strength of the friendship graph also holds the minimal irregularity strength, then the labeling is also an irregular labeling with the same strength as the modular case.

The connected graph G = (V, E) is classified as a Quadratic Embedding (QE) class based on the distance matrix D which is conditionally definite negative and if the quadratic embedding constant (QEC) of the graph G is non-positive. In this study, QEC be calculated for Squid Graph containing C3, C4, C5, and kite graph containing C4 . All of these graphs are included in the QE class, with QEC = 0 for squid graphs and kite graphs containing C4, QEC (Sq(3,m)) < 0 for squid graphs containing C3, and also for squid graphs containing C5.

A directed graph can be represented by several matrix representations, such as the anti-adjacency matrix. This paper discusses the general form of characteristic polynomial and eigenvaluesof the anti-adjacencymatrix of directed unicyclic corona graph. The characteristic polynomial of the anti-adjacency matrix can be found by counting the sum of the determinant of the anti-adjacency matrix of the directed cyclic inducedsubgraphs and the directed acyclic induced subgraphs from the graph. The eigenvalues of the anti-adjacency matrix can be real or complex numbers. We prove that the coefficient of the characteristic polynomial and the eigenvalues of the anti-adjacency matrix of directed unicyclic corona graph can be expressed in the function form that depends on the number of subgraphs contained in thedirected unicyclic corona graphs.

A directed cyclic graph is a directed graph that has at least one directed cycle graph, that is a cycle graph in which all edges are oriented, so that the direction passes through each vertex once, except the end of vertex. The directed unicyclic tadpole graph T m , n → is the graph created by concatenating a cycle C m → and a path P n → with an edge from any vertex of C m → to a pendant of P n → for integers m ≥ 3 and n ≥ 1. Antiadjacency matrix is a directed graph representation matrix based on whether or not there is a relation between one vertex and the others. This paper gives the general form of coefficients of characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic tadpole graph. To find the general form of coefficients of characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic tadpole graph we need to do the grouping of induced subgraphs into acyclic and cyclic and verify with related theorems. After that, the characteristic polynomial is factorized and the roots are calculatedto find its eigenvalues. The coefficients of the characteristic polynomial consist of three distinct values and the eigenvalues are divided into odd case and even case.

A u − ν rainbow path is a path that connects two vertices u and ν in a graph G and every edge in that path has a different color. A connected graph G is called a rainbow graph if there is a rainbow path for every pair of vertices in G. For any two vertices u and ν in G, a rainbow u − ν geodesic in G is the shortest rainbow u − ν path. The d-local strong rainbow number (lsrcd) is the smallest number of colors needed to color the edges of G such that any two vertices with distance at most d can be connected by a rainbow geodesic. Thus, the value of d is in the interval 1 < d < diam(G). In this paper, we show the lsrcd of prism graphs with d = 2 and d = 3, and the generalized formula of lsrcd for any value of d.

An injective function f from set of vertices in graph G to a set of {0,1,...,|E| − 1} is called an odd harmonious labeling if the function f induced the edge function f* from the set of edges of G to a set of odd positive integer number {1,3,5,...,2|E| − 1} with f*(xy) = f(x) + f(y) for every edge xy in E. Graph that has an odd harmonious labeling is called odd harmonious graph. The squid graph Tn,k is a graph which is obtained from a cycle Cn and we add k pendant to one vertex of the cycle. It is known that Cn is an odd harmonious graph if and only if n = 0 mod 4. However, by adding at least one pendant in the cycle graph, we can label the new graph odd harmoniously for all even number of vertices. In this paper, we showed that the graph Tn,k and T2n,k are an odd harmonious graph, for n = 0 (mod 2), n ≥ 4 and k ≥ 1. The construction of the odd harmonious labeling of the graph Tn,k and T2n,k are inspired by the odd harmonious labeling of Cn for n = 0(mod 4).

Secret sharing scheme is a technique to share secret data into n pieces based on a simple (k, n) threshold scheme. However, the problem with Shamir's secret sharing scheme is that they do not provide any way to verify that the dealer was honest and the shares were indeed valid. This problem also occurr in the Thien and Lin's image secret sharing or the Li Bai's construction using matrix projection. On the other hand, a developed protocol for secret sharing called verifiable secret sharing allows every participant to validate their received piece to confirm the authenticity of the secret. Therefore, this paper discussed a proposed scheme based on verifiable secret sharing, in which the matrix projection is used to create image shares and a public matrix from watermark image. The secrets were represented in elements of a square matrix. The watermark image was used for verifiability where the reconstructed watermark image verifies the accuracy of the reconstructed secret image.

One of the topics for decoding linear codes is an error-correcting pair. In this paper, it would be proved that the non-GRS MDS code and the AMDS [n, 1, n - 1] code have 1-error-correcting pair. It will also show that the AMDS code [n, n - 1,1] has no error-correcting pair.

Machine learning had been widely used to analyze various kinds of data, including sensitive data such as medical and financial data. A trained machine learning model can be wrapped in a web application so that people can access it easily via internet. However, if the data to be analyzed is private or confidential, this will cause a problem; the application administrator may read the input. As shown by Dowlin et al. in their remarkable paper, this kind of problem can be solved with homomorphic encryption scheme. Paillier encryption scheme is one kind of encryption scheme that has homomorphic property. In this research, we will show that one type of machine learning model can take an input encrypted by Paillier encryption scheme and produce an encrypted output that shares the same key. A machine learning model will be trained with the MNIST database of hand-written digits. This model will be tested with the test data encrypted with Paillier encryption scheme. The experiment shows that the model achieved 92.92% accuracy on the test set.