A graph H is d-degenerate if every subgraph of it contains a vertex of degree smaller than d. For a graph G, let ad(G) denote the maximum number of vertices of an induced d-degenerate subgraph of G. Sharp lowers bounds for %(G) in terms of the degree sequence of G are obtained, and the minimum number of edges of a graph G with n vertices and ~2(G) < m is determined precisely for all m < n.

The first geometric-arithmetic index was introduced in the chemical theory as the summation of 2 du dv /(du  dv ) overall edges of the graph, where du stand for the degree of the vertex u. In this paper we give the expressions for computing the first geometric-arithmetic index of hexagonal systems and phenylenes and present new method for describing hexagonal system by corresponding a simple g...

todeschini et al. have recently suggested to consider multiplicative variants of additive graphinvariants, which applied to the zagreb indices would lead to the multiplicative zagrebindices of a graph g, denoted by ( ) 1  g and ( ) 2  g , under the name first and secondmultiplicative zagreb index, respectively. these are define as  ( )21 ( ) ( )v v gg g d vand ( ) ( ) ( )( )2 g d v d v gu...

We solve three enumerative problems concerning the families of planar maps. More precisely, we establish algebraic equations for the generating function of loopless triangulations in which all vertices have degree at least d, for a certain value d chosen in {3, 4, 5}. The originality of the problem lies in the fact that degree restrictions are placed both on vertices and faces. Our proofs first...

Two permutations of the vertices of a graph G are called G-different if there exists an index i such that i-th entry of the two permutations form an edge in G. We bound or determine the maximum size of a family of pairwise G-different permutations for various graphs G. We show that for all balanced bipartite graphs G of order n with minimum degree n/2 − o(n), the maximum number of pairwise G-di...

Real-life networks often encounter vertex dysfunctions, which are usually followed by recoveries after appropriate maintenances. In this paper we present our research on a model of scale-free networks whose vertices are regularly removed and put back. Both the frequency and length of time of the disappearance of each vertex depend on the degree of the vertex, creating a heterogeneous control ov...

Given positive integers k m n, a graphG of order n is ðk;mÞ-pancyclic if for any set of k vertices of G and any integer r with m r n, there is a cycle of length r containing the k vertices. Minimum degree conditions and minimum sum of degree conditions of nonadjacent vertices that imply a graph is ðk;mÞ-pancylic are proved. If the additional property that the k vertices must appear on the cycle...

Let G be a triangle-free graph with n vertices and average degree t. We show that G contains at least e(1−n −1/12) 1 2 n t ln t( 1 2 ln t−1) independent sets. This improves a recent result of the first and third authors [8]. In particular, it implies that as n → ∞, every triangle-free graph on n vertices has at least e(c1−o(1)) √ n lnn independent sets, where c1 = √ ln 2/4 = 0.208138... Further...

‎The first variable Zagreb index of graph $G$ is defined as‎ ‎begin{eqnarray*}‎ ‎M_{1,lambda}(G)=sum_{vin V(G)}d(v)^{2lambda}‎, ‎end{eqnarray*}‎ ‎where $lambda$ is a real number and $d(v)$ is the degree of‎ ‎vertex $v$‎. ‎In this paper‎, ‎some upper and lower bounds for the distribution function and expected value of this index in random increasing trees (rec...

This paper generalises the concept of vertex pancyclic graphs. We define a graph as set-pancyclic if for every set S of vertices there is a cycle of every possible length containing S. We show that if the minimum degree of a graph exceeds half its order then the graph is set-pancyclic. We define a graph as k-ordered-pancyclic if, for every set S of cardinality k and every cyclic ordering of S, ...