7 research outputs found
Complete solution to a problem on the maximal energy of unicyclic bipartite graphs
The energy of a simple graph , denoted by , is defined as the sum of
the absolute values of all eigenvalues of its adjacency matrix. Denote by
the cycle, and the unicyclic graph obtained by connecting a vertex of
with a leaf of \,. Caporossi et al. conjecture that the
unicyclic graph with maximal energy is for and .
In``Y. Hou, I. Gutman and C. Woo, Unicyclic graphs with maximal energy, {\it
Linear Algebra Appl.} {\bf 356}(2002), 27--36", the authors proved that
is maximal within the class of the unicyclic bipartite -vertex
graphs differing from \,. And they also claimed that the energy of
and is quasi-order incomparable and left this as an open problem. In
this paper, by utilizing the Coulson integral formula and some knowledge of
real analysis, especially by employing certain combinatorial techniques, we
show that the energy of is greater than that of for
and , which completely solves this open problem and partially solves
the above conjecture.Comment: 8 page
Solution to a conjecture on the maximal energy of bipartite bicyclic graphs
The energy of a simple graph , denoted by , is defined as the sum of
the absolute values of all eigenvalues of its adjacency matrix. Let
denote the cycle of order and the graph obtained from joining
two cycles by a path with its two leaves. Let
denote the class of all bipartite bicyclic graphs but not the graph ,
which is obtained from joining two cycles and ( and ) by an edge. In [I. Gutman, D.
Vidovi\'{c}, Quest for molecular graphs with maximal energy: a computer
experiment, {\it J. Chem. Inf. Sci.} {\bf41}(2001), 1002--1005], Gutman and
Vidovi\'{c} conjectured that the bicyclic graph with maximal energy is
, for and . In [X. Li, J. Zhang, On bicyclic graphs
with maximal energy, {\it Linear Algebra Appl.} {\bf427}(2007), 87--98], Li and
Zhang showed that the conjecture is true for graphs in the class
. However, they could not determine which of the two graphs
and has the maximal value of energy. In [B. Furtula, S.
Radenkovi\'{c}, I. Gutman, Bicyclic molecular graphs with the greatest energy,
{\it J. Serb. Chem. Soc.} {\bf73(4)}(2008), 431--433], numerical computations
up to were reported, supporting the conjecture. So, it is still
necessary to have a mathematical proof to this conjecture. This paper is to
show that the energy of is larger than that of , which
proves the conjecture for bipartite bicyclic graphs. For non-bipartite bicyclic
graphs, the conjecture is still open.Comment: 9 page
Hardness Results for Total Rainbow Connection of Graphs
A total-colored path is total rainbow if both its edges and internal vertices have distinct colors. The total rainbow connection number of a connected graph G, denoted by trc(G), is the smallest number of colors that are needed in a total-coloring of G in order to make G total rainbow connected, that is, any two vertices of G are connected by a total rainbow path. In this paper, we study the computational complexity of total rainbow connection of graphs. We show that deciding whether a given total-coloring of a graph G makes it total rainbow connected is NP-Complete. We also prove that given a graph G, deciding whether trc(G) = 3 is NP-Complete
Hardness Results for Total Rainbow Connection of Graphs
A total-colored path is total rainbow if both its edges and internal vertices have distinct colors. The total rainbow connection number of a connected graph G, denoted by trc(G), is the smallest number of colors that are needed in a total-coloring of G in order to make G total rainbow connected, that is, any two vertices of G are connected by a total rainbow path. In this paper, we study the computational complexity of total rainbow connection of graphs. We show that deciding whether a given total-coloring of a graph G makes it total rainbow connected is NP-Complete. We also prove that given a graph G, deciding whether trc(G) = 3 is NP-Complete