# The minimum restricted edge-connected graph and the minimum size of graphs with a given edge–degree

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Abstract : Let $G=(V(G),E(G))$ be a graph. Determining the minimum and/or maximum size $(|E(G)|)$ of graphs with some given parameters is a classic extremal problem in graph theory. For a graph $G$ and $e=uv∈E(G)$, we denote $d(e)=d(u)+d(v)−2$ the edge–degree of $e$. In this paper, we obtain a lower bound for the minimum size of graphs with a given order $n$, a given minimum degree $\delta$ and a given minimum edge–degree $2\delta+k−2$. Moreover, we characterize the extremal graphs for $k=0,1,2$. As an application, we characterize some kinds of minimum restricted edge connected graphs.
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Journal articles
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https://hal.archives-ouvertes.fr/hal-01132310
Contributor : Hao Li <>
Submitted on : Tuesday, March 17, 2015 - 12:36:52 AM
Last modification on : Thursday, July 8, 2021 - 3:49:52 AM

### Citation

Weihua Yang, Yingzhi Tian, Hengzhe Li, Hao Li, Xiaofeng Guo. The minimum restricted edge-connected graph and the minimum size of graphs with a given edge–degree. Discrete Applied Mathematics, Elsevier, 2014, 167, pp.304-309. ⟨10.1016/j.dam.2013.10.028⟩. ⟨hal-01132310⟩

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