The sum of squares of degrees of bipartite graphs

• Published in: Acta mathematica Hungarica Vol. 171; no. 1; pp. 1 - 11
• Main Author: Neubauer, M. G.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-11-2023; Springer Nature B.V
• Subjects: Apexes; Graph theory; Graphs; Mathematics; Mathematics and Statistics; primary 05C07; 05C75; sum of squares of degree sequences; bipartite graph; 05C35; secondary 11P81
• ISSN: 0236-5294; 1588-2632
• DOI: 10.1007/s10474-023-01379-7

Let
G
be a subgraph of the complete bipartite graph
K
l
,
m
,
l
≤
m
, with
e
=
q
m
+
p
>
0
,
0
≤
p
<
m
, edges. The maximal value of the sum of the squares of the degrees of the vertices of
G
is
q
m
2
+
p
2
+
p
(
q
+
1
)
2
+
(
m
-
p
)
q
2
. We classify all graphs that attain this bound using the diagonal sequence of a partition.