TY - JOUR
T1 - On the integral weighted oriented unicyclic graphs with minimum skew energy
AU - Gong, Shi-Cai
AU - Hou, Yao-Ping
AU - Woo, Ching-Wah
AU - Xu, Guang-Hui
AU - Shen, Xiao-Ling
PY - 2013/6/1
Y1 - 2013/6/1
N2 - Let Gσ be a weighted oriented graph, which is obtained from a simple weighted undirected graph by assigning an orientation to each of its edges. For a (weighted) oriented graph Gσ, the undirected graph obtained from Gσ by removing the orientation and the weight of each of its arcs is called the underlying graph of G σ, denoted by Ĝ. By U(n,m)(m≥n) we denote the set of all connected integral weighted oriented unicyclic graphs with order n in which each arc is assigned a positive integral weight and the sum of the weights of all arcs is equal to the specified integer m. In this paper, we investigate the minimal skew energies of integral weighted unicyclic oriented graphs, showing that the underlying graph of the oriented graph with minimum skew energy among all graphs over U(n,m)(n≥6) is Sn,3, the graph obtained from a triangle by attaching n-3 pendent edges in exactly one of its vertices. Moreover, we show that its weight sequence has form(w1,a, a,.,aï̧·k,a+ 1,a+1,.,a+1ï̧·n-3-k,w2,w3)in which the arc lying on the cycle C3 and incident to no pendent arcs has weight w1, and the two largest weights correspond the other two arcs of C3, where positive integer numbers w1,w 2,w3,a and k satisfy 1≤w1≤a,a+1≤w 2≤w3≤a+w1 and m=w1+ka+(n-k-3) (a+1)+w2+w3. In addition, we determine the weight sequence above for n≤m≤3n-1.
AB - Let Gσ be a weighted oriented graph, which is obtained from a simple weighted undirected graph by assigning an orientation to each of its edges. For a (weighted) oriented graph Gσ, the undirected graph obtained from Gσ by removing the orientation and the weight of each of its arcs is called the underlying graph of G σ, denoted by Ĝ. By U(n,m)(m≥n) we denote the set of all connected integral weighted oriented unicyclic graphs with order n in which each arc is assigned a positive integral weight and the sum of the weights of all arcs is equal to the specified integer m. In this paper, we investigate the minimal skew energies of integral weighted unicyclic oriented graphs, showing that the underlying graph of the oriented graph with minimum skew energy among all graphs over U(n,m)(n≥6) is Sn,3, the graph obtained from a triangle by attaching n-3 pendent edges in exactly one of its vertices. Moreover, we show that its weight sequence has form(w1,a, a,.,aï̧·k,a+ 1,a+1,.,a+1ï̧·n-3-k,w2,w3)in which the arc lying on the cycle C3 and incident to no pendent arcs has weight w1, and the two largest weights correspond the other two arcs of C3, where positive integer numbers w1,w 2,w3,a and k satisfy 1≤w1≤a,a+1≤w 2≤w3≤a+w1 and m=w1+ka+(n-k-3) (a+1)+w2+w3. In addition, we determine the weight sequence above for n≤m≤3n-1.
KW - Integral weighted graph
KW - Oriented graph
KW - Skew energy
KW - Skew symmetric matrix
UR - http://www.scopus.com/inward/record.url?scp=84876671115&partnerID=8YFLogxK
UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-84876671115&origin=recordpage
U2 - 10.1016/j.laa.2013.02.018
DO - 10.1016/j.laa.2013.02.018
M3 - 21_Publication in refereed journal
VL - 439
SP - 262
EP - 272
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
SN - 0024-3795
IS - 1
ER -