The author studies the joint continuity (or conjoining property) of graph topologies \(\Gamma_1\) and \(\Gamma_2\) as well as an Ascoli type result for \(\Gamma_1\) in terms of even continuity. \(\Gamma_2\) was introduced by the reviewer [Trans. Am. Math. Soc. 123, 267--272 (1966; Zbl 0151.29703)] and \(\Gamma_1\) by the author [Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 15, 71--80 (1967; Zbl 0173.25002)] as well as by \textit{R. Arens} and \textit{J. Dugundji} [Pac. J. Math. 1, 5--31 (1951; Zbl 0044.11801)] as the finest closed-open topology. It is proved that both \(\Gamma_1\) and \(\Gamma_2\) are conjoining if either the domain \(X\) or the range \(Y\) is \(T_3\). An example is given where \(\Gamma_1\) is not conjoining. The author concludes with the Ascoli-type result: Let \(X\) be compact and \(Y\) \(T_3\). Let \(H \subset C(X,Y)\). Then \(H\) is \(\Gamma_1\)-compact if and only if \(H\) is \(\Gamma_1\)-closed, \(H(x)\) is compact for each \(x \in X\) and \(H\) is evenly continuous.