Ljiljana Pavlović

Maximal value of the zeroth-order Randić index

Let G(n,m) be a connected graph without loops and multiple edges which has n vertices and m edges. We find the graphs on which the zeroth-order connectivity index, equal to the sum of degrees of vertices of G(n,m) raised to the power -½, attains maximum.

Wiener Numbers of Phenylenes: An Exact Result

An exact relation is established between the Wiener number (W) of a phenylene (PH), its hexagonal squeeze (HS), and inner dual (ID):  W(PH) = (9/4)[W(HS) + 16W(ID) − (2h+1)(4h+1)], where h stands for the number of hexagons. Various applications of this relation are pointed out.

A search game on the union of graphs with immobile hider

We consider a search game for an immobile hider on one arc of the union of n graphs joined at one or two points. We evaluate a lower bound on the value of a strategy for the hider on this union. When we have identical graphs, we give the conditions under which the value of the strategy for the hider on this union is greater than or equal to n times the value of this strategy on one graph. We also solve search games on graphs, consisting of an odd number of arcs, each of length one, joining two points.

Extremal graphs for the geometric-arithmetic index with given minimum degree

Let G ( k , n ) be the set of connected simple n -vertex graphs with minimum vertex degree k . The geometric-arithmetic index G A ( G ) of a graph G is defined by? G A ( G ) = ? u v 2 d u d v d u + d v , where d ( u ) is the degree of vertex u and the summation extends over all edges u v of G . In this paper we find for k ? ? k 0 ? ,?with k 0 = q 0 ( n - 1 ) , where q 0 ? 0.088 is the unique positive root of the equation q q + q + 3 q - 1 = 0 , extremal graphs in G ( k , n ) for which the geometric-arithmetic index attains its minimum value, or we give a lower bound. We show that when k or n is even, the extremal graphs are regular graphs of degree k .

Extremal graphs for the Randić index when minimum, maximum degrees and order of graphs are odd

Let be the set of connected simple n-vertex graphs with minimum vertex degree and maximum vertex degree . The Randić index of a graph G is defined by , where is the degree of vertex u and the summation extends over all edges uv of G. In this paper, we find for , and k, m, n are odd, extremal graphs in for which the Randić index attains its minimum value. We show that the extremal graphs have vertices of degree k, m and , the number of vertices of degree is one and the number of vertices of degree k is as close to as possible.

Search for an infiltrator

We have solved a discrete search game on an array of n ordered cells for n ⩽ 9, with two players: infiltrator (hider) and searcher, who have opposite goals. The infiltrator wishes to reach the last cell number n (in finite time) and the searcher has to defend that cell. The payoff (to the hider) is the probability that the hider wins, that is, reaches the last cell without getting captured.