Consider a simple graph with no isolated edges and at most one isolated point. Assign edge a positive integer weight , and define the corresponding vertex labels
to be equal to the sum of weights of edges incident on . Then the irregularity strength of a graph (also denoted , e.g., Dinitz et al. 1992), is the smallest value
of
such that the vertex weights are all distinct (Chartrand et al. 1988, Przybylo
2024).
Define
for a graph with one or more isolated edges or more than one isolated
point.
For any graph
other than the triangle graph on vertices with finite irregularity strength,

(1)

(Nierhoff 2000, Przybylo 2024), with equality for star graphs
with
(Przybylo 2024).
For a connected graph on vertices, Chartrand et al. (1988) gave the lower
bound

(2)

where
is the number of vertices of vertex degree and is the maximum vertex
degree of .
For a regular graph of degree , this condition reduces to

(3)

(Przybylo 2024). Another bound attributed to Jacobson and Lehel defines

(4)

then

(5)

(Ebert et al. 1990, Dinitz et al. 1992, Bohman and Kravitz 2004). Note that some authors (e.g., Ebert et al. 1990, Bohman and Kravtiz 2004) leave
as the potentially rational number defined above, while others (e.g., Dinitz et
al. 1992) explicitly add a ceiling function
around the right hand side to make it an integer.
For a graph with minimum vertex degree ,

(6)

(Kalkowski et al. 2011, Przybylo 2024).
Faudree and Lehel (1987) conjectured that there exists an absolute constant such that for every regular graph with vertices and ,

(7)

(Przybylo 2024).
See also
Irregular Graph,
Regular
Graph
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Cite this as:
Weisstein, Eric W. "Irregularity Strength."
From MathWorldA Wolfram Web Resource. https://mathworld.wolfram.com/IrregularityStrength.html