70 research outputs found
On some interconnections between combinatorial optimization and extremal graph theory
The uniting feature of combinatorial optimization and extremal graph theory is that in both areas one should find extrema of a function defined in most cases on a finite set. While in combinatorial optimization the point is in developing efficient algorithms and heuristics for solving specified types of problems, the extremal graph theory deals with finding bounds for various graph invariants under some constraints and with constructing extremal graphs. We analyze by examples some interconnections and interactions of the two theories and propose some conclusions
The Minimal Total Irregularity of Graphs
In \cite{2012a}, Abdo and Dimitov defined the total irregularity of a graph
as
\hskip3.3cm
\noindent where denotes the vertex degree of a vertex . In
this paper, we investigate the minimal total irregularity of the connected
graphs, determine the minimal, the second minimal, the third minimal total
irregularity of trees, unicyclic graphs, bicyclic graphs on vertices, and
propose an open problem for further research.Comment: 13 pages, 4 figure
Non-regular graphs with minimal total irregularity
The {\it total irregularity} of a simple undirected graph is defined as
, where denotes the degree of a vertex .
Obviously, if and only if is regular. Here, we
characterize the non-regular graphs with minimal total irregularity and thereby
resolve the recent conjecture by Zhu, You and Yang~\cite{zyy-mtig-2014} about
the lower bound on the minimal total irregularity of non-regular connected
graphs. We show that the conjectured lower bound of is attained only if
non-regular connected graphs of even order are considered, while the sharp
lower bound of is attained by graphs of odd order. We also characterize
the non-regular graphs with the second and the third smallest total
irregularity
The total irregularity of a graph
In this note a new measure of irregularity of a graph G is introduced. It is
named the total irregularity of a graph and is defined as irrt(G) = 1 / 2∑u,v
∈V(G) |dG(u)-dG(v)|, where dG(u) denotes the degree of a vertex u ∈V(G). All
graphs with maximal total irregularity are determined. It is also shown that
among all trees of the same order the star has the maximal total irregularity
The Total Irregularity of Graphs under Graph Operations
The total irregularity of a graph is defined as \irr_t(G)=1/2 \sum_{u,v
\in V(G)} , where denotes the degree of a vertex . In this paper we give (sharp) upper bounds on the total irregularity
of graphs under several graph operations including join, lexicographic product,
Cartesian product, strong product, direct product, corona product, disjunction
and symmetric difference.Comment: 14 pages, 3 figures, Journal numbe
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