TY - JOUR
T1 - On path energy of graphs
AU - Akbari, Saieed
AU - Ghodrati, Amir Hossein
AU - Gutman, Ivan
AU - Hosseinzadeh, Mohammad Ali
AU - Konstantinova, Elena V.
PY - 2019/1/1
Y1 - 2019/1/1
N2 -
For a graph G with vertex set {v1, . . ., v
n
}, let P(G) be an n × n matrix whose (i, j)-entry is the maximum number of internally disjoint viv
j
-paths in G, if i ≠ j, and zero otherwise. The sum of absolute values of the eigenvalues of P(G) is called the path energy of G, denoted by PE. We prove that PE of a connected graph G of order n is at least 2(n− 1) and equality holds if and only if G is a tree. Also, we determine PE of a unicyclic graph of order n and girth k, showing that for every n, PE is an increasing function of k. Therefore, among unicyclic graphs of order n, the maximum and minimum PE-values are for k = n and k = 3, respectively. These results give affirmative answers to some conjectures proposed in MATCH.
AB -
For a graph G with vertex set {v1, . . ., v
n
}, let P(G) be an n × n matrix whose (i, j)-entry is the maximum number of internally disjoint viv
j
-paths in G, if i ≠ j, and zero otherwise. The sum of absolute values of the eigenvalues of P(G) is called the path energy of G, denoted by PE. We prove that PE of a connected graph G of order n is at least 2(n− 1) and equality holds if and only if G is a tree. Also, we determine PE of a unicyclic graph of order n and girth k, showing that for every n, PE is an increasing function of k. Therefore, among unicyclic graphs of order n, the maximum and minimum PE-values are for k = n and k = 3, respectively. These results give affirmative answers to some conjectures proposed in MATCH.
UR - http://www.scopus.com/inward/record.url?scp=85064385519&partnerID=8YFLogxK
M3 - Article
AN - SCOPUS:85064385519
VL - 81
SP - 465
EP - 470
JO - Match
JF - Match
SN - 0340-6253
IS - 2
ER -