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An injective function f from the set of vertices in a graph G to a set {0,1,...,n} is called graceful labeling if the function f induced the edge function f* from the set of edges of G to a set of positive integers {1,2,...,n} with f*(xy) = |f(x) − f(y)| for every edge xy ∈ E(G) such that every edge has a different label. A graph G that can be labeled with graceful labeling is called a graceful graph. The multiple-fan graph with k pendants F(n1,n2,...,nr),k is a graph obtained from r fan graphs Fni, with i = 1,2,...,r, which share the center vertex and then we add k pendants to the center vertex of the multiple-fan graph. In this paper, we show that the multiple-fan graph with k pendants is graceful, with ni ≥ 2, i = 1,2,..., r and k ≥ r − 1. [ABSTRACT FROM AUTHOR]

Consider G and H as simple graphs. A graph G has an H-covering if every edge of G belongs to a subgraph of G which the subgraph is isomorphic to H. The graph G is said to be H-magic if there exists a total labeling of G such that for every subgraph H′ of G where H′ is isomorphic to H, then the sum of all labels in H′ is constant. In this paper, we study grid-like graphs G, whose faces in F(G) are labeled. The magic labeling of type (1,1,1) means that the label is assigned to all vertices, edges, and faces, and the weight is calculated as ∑v∈V(G) l(v)+∑e∈E(G) l(e)+∑f∈F(G) l(f) with all weight for every face being constant. We show that certain unions of grid graphs are H-magic, where H is a grid graph. [ABSTRACT FROM AUTHOR]

A rainbow path in a graph G is an edge colored graph such that each edge in the path does not have the same color. The minimal number of colors which is needed to color the edges in the path so that every pair of vertices is connected by at least one rainbow path is called the rainbow connection number of G, denoted by rc(G). A rainbow path of length d(u, v), where d(u, v) is the distance between u and v, is called rainbow u - v geodesic in G. A strongly rainbow connected graph is a graph such that there exists a rainbow u − v geodesic for any two vertices u and v in G. The strong rainbow connection number src(G) of G is the minimum number of colors needed to make G strongly rainbow connected graph. In this paper, we prove the exact values of rc and src for crystal graph and its generalization to neuron graph. [ABSTRACT FROM AUTHOR]

Supposed G be a simple graph with a set of vertices (G), set of edges E(G), with | E(G)| = q. A graph G is a harmonious graph if there is an injective function f*: (G) → ℤq, such that the induced function f*: E (G) → ℤq defined by f*(xy) = f(x) + f(y), ∀xy ∈ E(G) is a bijective function. The function f is called harmonious labeling of G. It was known that a complete graph Kn is harmonious only for n ≤ 4 In this paper, we investigate the existence of harmonious labeling of the graphs from the corona operation between a complete graph K4 and Kn k̅n, and also between K5 and k̅n. [ABSTRACT FROM AUTHOR]

Clustering can be done by various methods, one of them is a graph clustering. One algorithm that can be used for graph clustering is based on the minimum spanning tree. In this algorithm, we count the value of the standard deviation weights of the vertices between nodes, which why it is called maximum standard deviation reduction (MSDR) algorithm. By using the MSDR algorithm, a cluster of optimal cluster results can be found, and the benefit is we do not have to determine the number of clusters. This method is called unsupervised learning. In MSDR algorithm, the number of clusters will be determined automatically by using polynomial regression. However, in a complex case, it will be difficult to obtain the number of clusters, so that a modification of the algorithm MSDR is proposed and called by MMSDR (Modification of Maximum Standard Deviation Reduction). The modification is done by replacing the polynomial regression to calculate the difference of standard deviation change in the value of the previous calculation. This paper will discuss the results of the implementation of the algorithm MMSDR on the data that come from Indonesia flights by airline X. While the input matrix in the MMSDR is using adjacency matrix, in this work we use matrix complement as the input. [ABSTRACT FROM AUTHOR]

Maximum Standard Deviation Reduction (MSDR) is a graph clustering algorithm to minimize the distance variation within a cluster. In this paper we propose a modified MSDR by replacing one technical step in MSDR which uses polynomial regression, with a new and simpler step. This leads to our new algorithm called Modified MSDR (MMSDR). We implement the new algorithm to separate a domestic flight network of an Indonesian airline into two large clusters. Further analysis allows us to discover a weak link in the network, which should be improved by adding more flights. [ABSTRACT FROM AUTHOR]

Maximal clique enumeration is a graph clustering method for finding all vertices that have the most influence in a graph. The Bron-Kerbosch algorithm is one of the fastest algorithms to find all maximal cliques. Hence, this paper will focus on that algorithm to find all maximal cliques. Counting all maximal cliques of a graph usually uses an adjacency matrix of the graph to find all vertices in the graph that have the most influence. But, in this paper, a complement matrix of a graph will be used in counting all maximal cliques. This research uses a graph that represents a domestic flight route of one of the airlines in Indonesia. By using Bron-Kerbosch algorithm, all maximal cliques of the graph will be listed, so that it will come up with the vertices which are the most influential in this graph. The graph will be represented in complement matrix. The result of applying the Bron-Kerbosch algorithm with the complement matrix of graph will determine vertices that have the most influence in the graph. Besides that, by using a complement matrix, the result gives more information on the vertices which are only connected to the vertices that have the most influence. [ABSTRACT FROM AUTHOR]