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G(V, E) represents the basic chart without circle, if existing a one-to-one mapping f:E(G) → {1,2,...,|E|}, for any two vertices in the diagram, in the event that d(u)= d(v). S(u)=S(v), where S(u)= Σuw∈E(G)f(uw), d(u) represents the degree of the vertex u, then call the mapping f: Adjacent Vertex Reducible Edge Labeling (alluded as AVREL). In graph theory, graph coloring and graph labeling are two research directions of graph theory, and there is little correlation between the two in previous research results. In the process of researching the concept of Adjacent Reducible Edge Coloring proposed by Professor Zhang Zhongfu, we found that there are several graph classes whose coloring number reaches the sum of the number of vertices and edges, so we propose a new concept of Adjacent Reducible Edge Labeling. In the transportation network, the edge weight represents the transportation capacity, and the node transportation capacity is represented by the sum of its associated edges. Two nodes with the same degree of adjacency require the transportation capacity to be as equal as possible, which can be described by the Adjacent Vertex Reducible Edge Coloring model. when the road diversity reaches the extreme value, it can be described by the Adjacent Vertex Reducible Edge Labeling model. In this paper, designing and using Adjacent Vertex Reducible Edge Labeling algorithm (abbreviation: AVREL algorithm). The algorithm recursively looks through the arrangement space of the Adjacent Reducible Edge Label through the underlying label of the edge, lastly sifts through the graph book fulfilling the edge label and results as a label matrix. In the wake of examining the algorithm results, some special graphs such as Petersen-pyramid graphs, Möbius ladder graphs, bicyclic graphs, and some joint graphs in various situations are summed up, the proofs and conjectures are given. [ABSTRACT FROM AUTHOR]

A graph G(p, q) is said to have an edge-magic total labeling if there exists a bijective function f: V(G) ∪ E(G) → {1, 2, ...,p + q}, such that for any edge uv∊E(G) the condition f(u)+f(v)+f(uv)=k is satisfied, k is a constant. In this paper, a new algorithm, based on the graph generate algorithm, is designed to obtain the edge-magic total labeling of the unicyclic graphs. Some theorems about unicyclic graphs are also deduced from the algorithm's results. It's believed that the algorithm proposed is innovative and can be adopted by other researchers. [ABSTRACT FROM AUTHOR]