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Here, we propose a new kind of Estrada index based on the Gaussianization of the generalized distance matrix of a graph. The generalized distance matrix D α ( G ) is signified by D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ), where α ∈. As discussed earlier that even-even language can be expressed by a regular expression (aa+bb+(ab+ba)(aa+bb)*(ab+ba))* The language even-even may be accepted by the following GTG.For a simple undirected connected graph G of order n, let D ( G ), D L ( G ), D Q ( G ) and T r ( G ) be, respectively, the distance matrix, the distance Laplacian matrix, the distance signless Laplacian matrix and the diagonal matrix of the vertex transmissions of G. Figure 2 : Generalized Transition Graph (GTG) ExampleĬonsider the even-even language, defined over Σ =. Note: There is no difference between the Kleene star (*) closure for regular expressions and a loop in transition graphs, as illustrated in the following figure. The first edge should be labeled (ba + a) as in the figure above, not (ab + a). Note that the word containing the single letter b can take the free ride along the ∧-edge from start to middle, and then have letter b read to reach to the final state. This GTG accepts all strings without a double b. It may be noted that in GTG, the labels of transition edges are corresponding regular expressions.Įxample: Figure 1 : Generalized Transition Graph (GTG) Example Directed edges in GTG connecting some pair of states labeled with regular expression.GTG consist finite set of input letters (Σ) from which input strings are formed.GTG consists finite number of states, at least one of which is start state and some (maybe none) final states.Informally a generalized transition graph is a collection of three things are as follows A finite set, δ of transitions, (directed edge labels) (u, s, v), where u, v ∈ Q and s is a regular expression over Σ.F : A set of final or accepting states F ⊆.S : A non-empty set set of start states, S ⊆.
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