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Let G is a connected graph. A rainbow k-coloring on G is a function c : E(G) → {1, …, k for k ∈ ℕ where for any two vertices u and v in V, there is a path which all edges have no same color. A path that all edges have no the same color is called a rainbow path. Let k is the smallest positive integer that is needed to make G be rainbow connected, then k is called by the rainbow connection number of G symbolized by rc(G). There are many researches on rainbow connection number of operation graph classes that have been done. In this paper, we use join product and strong product as operations on sun graph cn⊙ K̅1 and path graph Pm for determining the rainbow connection number. [ABSTRACT FROM AUTHOR]

An adjacency matrix A(G) of directed graph G is an m×m matrix consisting of only entries 0 and 1, where m is the number of vertices of G. The entry aij is equal to 1 if there exists a directed edge from vertex vi to vertex vj, otherwise it is equal to 0. Let D(G) be a diagonal matrix of size m×m with each of its main diagonal entry being the degree of the corresponding vertex of directed graph G. Then the matrix L(G) = D(G) − A(G) is called the Laplacian matrix of G. Since a directed graph has two types of degrees namely indegree and outdegree, they result in directed graphs also having the both types of its Laplacian matrices. In this study, the adjacency matrix and the Laplacian matrix of cyclic directed prism graph are investigated. The general form of the coefficients of polynomial characteristic of the Adjacency matrix and Laplacian matrix, respectively is obtained by applying the row reduction method in linear algebra, whereas the general form of the eigenvalues of the polynomial characteristic of the Adjacency matrix and Laplacian matrix, respectively is obtained by factorization and substitution methods. [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]

Let F, G, and H be simple graphs. We say F → (G,H) if for every 2-coloring of the edges of F there exist a monochromatic G or H in F. The Ramsey number r(G,H) is min {v(F): F → (G,H)}, the size Ramsey number r* (G,H) is min {e(F) : F → (G,H)}, and the restricted size Ramsey number r* (G,H) is min {e(F) : F → (G,H), (F) = r(G,H)}. In 1972, Chvátal and Harary gave the Ramsey number for P3 versus any graph H with no isolates. In 1983, Harary and Miller started the investigation of the (restricted) size Ramsey number for some pairs of small graphs with order at most four. In1983, Faudree and Sheehan continued the investigation and summarized the complete results on the (restricted) size Ramsey number for all pairs of small graphs with order at most four. In 1998, Lortz and Mengenser gave both the size Ramsey number and the restricted size Ramsey number for all pairs of small forests with order at most five. Lately, we investigate the restricted size Ramsey number for a path of order three versus all connected graphs of order five. In this work, we continue the investigation on the restricted size Ramsey number for a pair of small graphs. In particularly, for a path of order three versus connected graphs of order six. [ABSTRACT FROM AUTHOR]