id: 06015780
dt: j
an: 06015780
au: Abdo, Hosam; Dimitrov, Darko; Gutman, Ivan
ti: On the Zagreb indices equality.
so: Discrete Appl. Math. 160, No. 1-2, 1-8 (2012).
py: 2012
pu: Elsevier Science B.V. (North-Holland), Amsterdam
la: EN
cc: 
ut: first Zagreb index; second Zagreb index; comparing Zagreb indices
ci: 
li: doi:10.1016/j.dam.2011.10.003
ab: Summary: For a simple graph $G=(V,E)$ with $n$ vertices and $m$ edges, the
    first Zagreb index and the second Zagreb index are defined as $M_{1}(G)
    = \sum_{v\in V}d(v)^{2}$ and $M_{2}(G) = \sum_{uv\in E}d(u)d(v)$, where
    $d(u)$ is the degree of a vertex $u$ of $G$. In [{\it D. Vukičević},
    {\it I. Gutman}, {\it B. Furtula}, {\it V. Andova} and {\it D.
    Dimitrov}, “Some observations on comparing Zagreb indices,” MATCH
    Commun. Math. Comput. Chem. 66, No. 2, 627‒645 (2011)], it was shown
    that if a connected graph $G$ has maximal degree 4, then $G$ satisfies
    $M_{1}(G)/n=M_{2}(G)/m$ (also known as the Zagreb indices equality) if
    and only if $G$ is regular or biregular of class 1 (a biregular graph
    for which no two vertices of same degree are adjacent). There, it was
    also shown that there exist infinitely many connected graphs of maximal
    degree $Δ=5$ that are neither regular nor biregular of class 1 which
    satisfy the Zagreb indices equality. Here, we generalize that result by
    showing that there exist infinitely many connected graphs of maximal
    degree $Δ\geq 5$ that are neither regular nor biregular graphs of
    class 1 which satisfy the Zagreb indices equality. We also consider the
    cases where the above equality holds when the degrees of vertices of a
    given graph are in a prescribed interval of integers.
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