For a graph \(F\), the Tur\'an number \(\ex(n,F)\) is the maximum number of edges in an \(F\)-free graph on \(n\) vertices. Let \(q\ge 2\) be an integer and set \(n=q^{2}+q+1\). Brown and Erd\H{o}s, R\'enyi and S\'os independently proved that $\ex(n,C_{4})\ge \frac12 q(q+1)^{2}$ for every prime power \(q\), and F\"ured...
The number of spanning trees is a classical graph invariant and an important measure of network reliability, as it counts the minimal connected spanning substructures that can maintain communication in a network. Let $\mathcal{B}(n,d)$ be the set of connected bipartite graphs of order $n$ and diameter $d$. Motivated by...
Shao-Han Xu, Ivan Damnjanović, Ke-Xiang Xu· 0 citations
The transmission of a vertex $v$ in a connected graph $G$ is the sum of distances from $v$ to all vertices in $G$. A transmission irregular (TI) graph is a connected graph in which any two distinct vertices have different transmissions. We extend the concept of transmission to edges by defining the transmission of an e...
Kexiang Xu, Ivan Damnjanović, Urovs Milivojevi'c et al.· arXiv.org· 0 citations
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