Geodesic
Note on Irregular Graphs
Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 43 (2018), p. 5 .

Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

Let $G$ be a graph with vertex set $\mathbf V(G)$ and edge set $\mathbf E(G)$. For $v \in \mathbf V(G)$, by $d_G(v)$ is denoted the degree of the vertex $v$. A graph in which not all vertices have equal degrees is said to be irregular. Different quantitative measures of irregularity have been proposed, of which the Albertson index $irr(G) = \sum_{uv \in \mathbf E(G)} |d_G(u)-d_G(v)|$ is the most popular. We compare $irr(G)$ with the recently introduced sigma-index $\sigma(G) = \sum_{uv \in \mathbf E(G)} [d_G(u)-d_G(v)]^2$ and show that in the general case these are incomparable. Graphs in which $|d_G(u)-d_G(v)|=1$ holds for all $uv \in \mathbf E(G)$ are called stepwise irregular $($SI$)$. Several methods for constructing SI graphs are described.
Classification : 05C07, 05C35
Keywords: degree (of vertex), irregularity (of graph), stepwise irregular graph, Albertson index, $\sigma$ index 
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     author = {Ivan Gutman and Tam\'as R\'eti},
     title = {Note on {Irregular} {Graphs}},
     journal = {Bulletin de l'Acad\'emie serbe des sciences. Classe des sciences math\'ematiques et naturelles},
     pages = {5 },
     publisher = {mathdoc},
     volume = {43},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BASS_2018_43_a1/}
}
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Ivan Gutman; Tamás Réti. Note on Irregular Graphs. Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 43 (2018), p. 5 . http://geodesic.mathdoc.fr/item/BASS_2018_43_a1/