Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 9 3 2024 09 01 1-Edge contraction: Total vertex stress and confluence number 527 538 14535 10.22049/cco.2023.27338.1238 EN Shiny Joseph Mathematic Research Center, Mary Matha Arts and Science College, Mananthavady, Kerala, India Johan Kok Independant Mathematics Researcher, City of Tshwane, South Africa & Visiting Faculty at CHRIST (Deemed to be a University), Bangalore, India Journal Article 2021 07 19 This paper introduces certain relations between $1$-edge contraction and the total vertex stress and the confluence number of a graph. A main result states that if a graph $G$ with $\zeta(G)=k\geq 2$ has an edge $v_iv_j$ and a $\zeta$-set $\mathcal{C}_G$ such that $v_i,v_j\in \mathcal{C}_G$ then, $\zeta(G/v_iv_j) = k-1$. In general, either $\mathcal{S}(G/e_i) \leq \mathcal{S}(G/e_j)$ or $\mathcal{S}(G/e_j) \leq \mathcal{S}(G/e_i)$ is true. This observation leads to an investigation into the question: for which edge(s) $e_i$ will $\mathcal{S}(G/e_i) = \max\{\mathcal{S}(G/e_j):e_j \in E(G)\}$ and for which edge(s) will $\mathcal{S}(G/e_j) = \min\{\mathcal{S}(G/e_\ell):e_\ell \in E(G)\}$? https://comb-opt.azaruniv.ac.ir/article_14535_7872a0fe9b978460e43157600b4820c7.pdf