One of the basic results in graph theory is Dirac’s theorem, that every graph of order n 3 and minimum degree n/2 is Hamiltonian. This may be restated as: if a graph of order n and minimum degree n/2 contains a cycleC then it contains a spanning cycle, which is just a spanning subdivision of C. We show that the same conclusion is true if instead of C, we choose any graph H such that every conne...

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...

Let G be a simple graph. For any k ∈ N , the k−power of G is a simple graph G with vertex set V (G) and edge set {xy : dG(x, y) ≤ k} and the k−subdivision of G is a simple graph G 1 k , which is constructed by replacing each edge of G with a path of length k. So we can introduce the m−power of the n−subdivision of G, as a fractional power of G, that is denoted by G m n . In other words G m

For any k ∈ N, the k−subdivision of 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 this paper we introduce the mth power of the n−subdivision of G, as a fractional power of G, denoted by G m n . In this regard, we investigate chromatic number and clique number of fractional power of graphs. Also, we conjecture that χ(G m n ) = ω(G ...

Inspired by the chemical applications of higher-order connectivity index (or Randic index), we consider here the higher-order first Zagreb index of a molecular graph. In this paper, we study the linear regression analysis of the second order first Zagreb index with the entropy and acentric factor of an octane isomers. The linear model, based on the second order first Zag...

Given any integers s, t ≥ 2, we show there exists some c = c(s, t) > 0 such that any Ks,t-free graph with average degree d contains a subdivision of a clique with at least cd 1 2 s s−1 vertices. In particular, when s = 2 this resolves in a strong sense the conjecture of Mader in 1999 that every C4-free graph has a subdivision of a clique with order linear in the average degree of the original g...

For a graph G and a unary graph operation X, there is a graph sequence {Gk} generated by G0 = G and Gk+1 = X(Gk). Let Sp(Gk) denote the set of normalized Laplacian eigenvalues of Gk. The set of limit points of ⋃∞ k=0 Sp(Gk), lim infk→∞ Sp(Gk) and lim supk→∞ Sp(Gk) are considered in this paper for graph sequences generated by two operations: subdivision and triangle extension. It is obtained tha...

In this paper we develop a new description for subdivision surfaces based on a graph grammar formalism. Subdivision schemes are specified by a context sensitive grammar in which production rules represent topological and geometrical transformations to the surface’s control mesh. This methodology can be used for all known subdivision surface schemes. Moreover, it gives an effective representatio...

In the modern era, mathematical modeling consisting of graph theoretic parameters or invariants applied to solve problems existing in various disciplines physical sciences like computer sciences, physics, and chemistry. Topological indices (TIs) are one which frequently used identify different physicochemical structural properties molecular graphs. Wiener index is first distance-based TI that c...

We present two short proofs for Diestel's criterion that a connected graph has normal spanning tree provided it contains no subdivision of countable clique in which every edge been replaced by uncountably many parallel edges.