57,522 research outputs found
On the variable inverse sum deg index
Several important topological indices studied in mathematical chemistry are expressed in the following way Puv∈E(G) F(du, dv), where F is a two variable function that satisfies the condition F(x, y) = F(y, x), uv denotes an edge of the graph G and du is the degree of the vertex u. Among them, the variable inverse sum deg index ISDa, with F(du, dv) = 1/(dua + dva), was found to have several applications. In this paper, we solve some problems posed by Vukičević [1], and we characterize graphs with maximum and minimum values of the ISDa index, for a < 0, in the following sets of graphs with n vertices: graphs with fixed minimum degree, connected graphs with fixed minimum degree, graphs with fixed maximum degree, and connected graphs with fixed maximum degree. Also, we performed a QSPR analysis to test the predictive power of this index for some physicochemical properties of polyaromatic hydrocarbon
Rainbow domination and related problems on some classes of perfect graphs
Let and let be a graph. A function is a rainbow function if, for every vertex with
, . The rainbow domination number
is the minimum of over all rainbow
functions. We investigate the rainbow domination problem for some classes of
perfect graphs
Revan-degree indices on random graphs
Given a simple connected non-directed graph , we consider two
families of graph invariants:
(which has gained interest recently) and (that we introduce in this work); where denotes the edge of
connecting the vertices and , is the Revan degree of the
vertex , and is a function of the Revan vertex degrees. Here, with and the maximum and minimum
degrees among the vertices of and is the degree of the vertex .
Particularly, we apply both and R on two models of
random graphs: Erd\"os-R\'enyi graphs and random geometric graphs. By a
thorough computational study we show that \left and
\left, normalized to the order of the graph, scale
with the average Revan degree \left; here \left
denotes the average over an ensemble of random graphs. Moreover, we provide
analytical expressions for several graph invariants of both families in the
dense graph limit.Comment: 16 pages, 10 figure
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