The principle aim of this research article is to investigate the properties of k-fractional integration introduced and defined by Mubeen and Habibullah [1], and secondly to solve the integral equation of the form             1 1 0 , ; d , ; k x k k x t g x F t x f t k                t t 0, 0, 0,0 k x , for          , where     1 1 , ; , ; k F x k     ...

A graph G is called a fractional (k, n′,m)-critical deleted graph if any n′ vertices are removed from G the resulting graph is a fractional (k,m)-deleted graph. In this paper, we determine that for integers k ≥ 1, i ≥ 2, n′,m ≥ 0, n > 4ki+ n′ + 4m− 4, and δ(G) ≥ k(i− 1) + n′ + 2m, if |NG(x1) ∪NG(x2) ∪ · · · ∪NG(xi)| ≥ n+ n′ 2 for any independent subset {x1, x2, . . . , xi} of V (G), then G is a...

A graph G is called a fractional (k, n′,m)-critical deleted graph if any n′ vertices are removed from G the resulting graph is a fractional (k,m)-deleted graph. In this paper, we prove that for integers k ≥ 2, n′,m ≥ 0, n ≥ 8k + n′ + 4m− 7, and δ(G) ≥ k + n′ +m, if |NG(x) ∪NG(y)| ≥ n+ n′ 2 for each pair of non-adjacent vertices x, y of G, then G is a fractional (k, n′,m)-critical deleted graph....

‎The distinguishing number (resp. index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$‎ ‎such that $G$ has an vertex labeling (resp. edge labeling) with $d$ labels that is preserved only by a trivial‎ ‎automorphism‎. ‎For any $n in mathbb{N}$‎, ‎the $n$-subdivision of $G$ is a simple graph $G^{frac{1}{n}}$ which is constructed by replacing each edge of $G$ with a path of length $n$...

For any k ∈ N, the k-subdivision of a graph G is a simple graph G 1 k , which is constructed by replacing each edge of G with a path of length k. In [Moharram N. Iradmusa, On colorings of graph fractional powers, Discrete Math., (310) 2010, No. 10-11, 1551-1556] the mth power of the n-subdivision of G has been introduced as a fractional power of G, denoted by G m n . In this regard, we investig...

Fractional superstrings in the tensor-product formulation experience “internal projections” which reduce their effective central charges. Simple expressions for the characters of the resulting effective worldsheet theory are found. All states in the effective theory can be consistently assigned definite spacetime statistics. The projection to the effective theory is shown to be described by the...

and Applied Analysis 3 fractional order to distributed order fractional. In Section 4, we introduce the distributed order fractional evolution systems C doD α t x t A C doD β t x t Bu t , x 0 x0, 0 < β < α ≤ 1, 1.5 where u t is control vector, and generalize the results obtained in Section 3 for this case. Finally, the conclusions are given in the last section. 2. Elementary Definitions and The...