59 research outputs found
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 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
Spectral radius minus average degree: a better bound
Collatz and Sinogowitz had proposed to measure the departure of a graph
from regularity by the difference of the (adjacency) spectral radius and the
average degree: . We give here new lower
bounds on this quantity, which improve upon the currently known ones
Edges in Fibonacci cubes, Lucas cubes and complements
The Fibonacci cube of dimension n, denoted as , is the subgraph of
the hypercube induced by vertices with no consecutive 1's. The irregularity of
a graph G is the sum of |d(x)-d(y)| over all edges {x,y} of G. In two recent
paper based on the recursive structure of it is proved that the
irregularity of and are two times the number of edges
of and times the number of vertices of ,
respectively. Using an interpretation of the irregularity in terms of couples
of incident edges of a special kind (Figure 2) we give a bijective proof of
both results. For these two graphs we deduce also a constant time algorithm for
computing the imbalance of an edge. In the last section using the same approach
we determine the number of edges and the sequence of degrees of the cube
complement of
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