Tadpole domination in duplicated graphs

In this paper, selected domination parameters are discussed and proved especially after expanding the graph by duplicating vertices or edges. The tadpole domination number after expansion is obtained in terms of the old tadpole domination number and the maximal path of the original graph, tadpole domination number was determined for any graph after the duplication of the order of the vertices set of G or the duplication of the size of the edges set. Determining if a graph is Hamiltonian is much more problematic. Therefore, the leading outcome in this paper is that we provide that if an expanded graph (by duplicating each vertex by an edge) has tadpole domination, then the original (initial) graph has Hamiltonian path and vice versa.

Connective Eccentric Index of an Infinite Family of Linear Polycene Parallelogram Benzenoid

Let G=(V, E) be a graph, where V(G) is a non-empty set of vertices and E(G) is a set of edges.We defined dv denote the degree of vertex v∈V(G). The Eccentric Connectivity index ξ(G) and theConnective Eccentric index Cξ(G) of graph G are defined as ξ(G)= ∑ v∈V(G)dv x ξ(v) and Cξ(G)=ξ(G)= ∑ v∈V(G)dv x ξ(v)- where ε(v) is defined as the length of a maximal path connecting a vertex v toanother vertex of G. In this present paper, we compute these Eccentric indices for an infinite family oflinear polycene parallelogram benzenod by a new method.Keywords: Molecular graphs; Linear polycene parallelogram; Benzenoid; Eccentric connectivityindex; Connective eccentric index

An Analysis of the Height of Tries with Random Weights on the Edges

We analyse the weighted height of random tries built from independent strings of i.i.d. symbols on the finite alphabet {1, . . .d}. The edges receive random weights whose distribution depends upon the number of strings that visit that edge. Such a model covers the hybrid tries of de la Briandais and the TST of Bentley and Sedgewick, where the search time for a string can be decomposed as a sum of processing times for each symbol in the string. Our weighted trie model also permits one to study maximal path imbalance. In all cases, the weighted height is shown to be asymptotic toclognin probability, wherecis determined by the behaviour of thecoreof the trie (the part where all nodes have a full set of children) and the fringe of the trie (the part of the trie where nodes have only one child and formspaghetti-like trees). It can be found by maximizing a function that is related to the Cramér exponent of the distribution of the edge weights.

Modified Eccentric Connectivity Polynomial of Circumcoronene Series of Benzenoid Hk

Let G=(V,E) be a molecular graph, where V(G) is a non-empty set of vertices/atoms and E(G) is a set of edges/bonds. For vV(G), defined dv be degree of vertex/atom v and S(v)is the sum of the degrees of its neighborhoods. The modified eccentricity connectivity polynomial of a molecular graph G is defined as  where ε(v) is defined as the length of a maximal path connecting v to another vertex of molecular graph G. In this paper we compute this polynomial for a famous molecular graph of Benzenoid family.