17 research outputs found
Note on PI and Szeged indices
In theoretical chemistry molecular structure descriptors are used for
modeling physico-chemical, pharmacological, toxicologic, biological and other
properties of chemical compounds. In this paper we study distance-based graph
invariants and present some improved and corrected sharp inequalities for PI,
vertex PI, Szeged and edge Szeged topological indices, involving the number of
vertices and edges, the diameter, the number of triangles and the Zagreb
indices. In addition, we give a complete characterization of the extremal
graphs.Comment: 10 pages, 3 figure
On the extremal properties of the average eccentricity
The eccentricity of a vertex is the maximum distance from it to another
vertex and the average eccentricity of a graph is the mean value
of eccentricities of all vertices of . The average eccentricity is deeply
connected with a topological descriptor called the eccentric connectivity
index, defined as a sum of products of vertex degrees and eccentricities. In
this paper we analyze extremal properties of the average eccentricity,
introducing two graph transformations that increase or decrease .
Furthermore, we resolve four conjectures, obtained by the system AutoGraphiX,
about the average eccentricity and other graph parameters (the clique number,
the Randi\' c index and the independence number), refute one AutoGraphiX
conjecture about the average eccentricity and the minimum vertex degree and
correct one AutoGraphiX conjecture about the domination number.Comment: 15 pages, 3 figure
2-Edge Distance-Balanced Graphs
In a graph A, for each two arbitrary vertices g, h with
d(g,h)=2,|MAg2h|=mAg2h is introduced the number of edges of A that are closer
to g than to h. We say A is a 2-edge distance-balanced graph if we have
mAg2h=mAh2g. In this article, we verify the concept of these graphs and present
a method to recognize k-edge distance-balanced graphs for k = 2,3 using
existence of either even or odd cycles. Moreover, we investigate situations
under which the Cartesian and lexicographic products lead to 2-edge distance
-balanced graphs. In some subdivision-related graphs 2-edge distance-balanced
property is verified
Relationship between edge Szeged and edge Wiener indices of graphs
Let G be a connected graph and ξ(G) = Sze(G) - We(G), where We(G) denotes the edge Wiener index and Sze(G) denotes the edge Szeged index of G. In an earlier paper, it is proved that if T is a tree then Sze(T) = We(T). In this paper, we continue our work to prove that for every connected graph G, Sze(G) ≥ We(G) with equality if and only if G is a tree. We also classify all graphs with ξ(G) ≤ 5. Finally, for each non-negative integer n ≠ 1 there exists a graph G such that ξ(G) = n
On distance-balanced generalized Petersen graphs
A connected graph of diameter is
-distance-balanced if for every with
, where is the set of vertices of that are closer
to than to . We prove that the generalized Petersen graph is
-distance-balanced provided that is large enough
relative to . This partially solves a conjecture posed by Miklavi\v{c} and
\v{S}parl \cite{Miklavic:2018}. We also determine when
is large enough relative to