Extremal values of eccentric resistance distance sums of graphs in certain classes

Let 𝐺 be a connected graph with the vertex set 𝑉 𝐺 and let 𝑟 𝐺 ⁡ ( 𝑥 , 𝑦 ) be the resistance distance between the vertices 𝑥 and 𝑦 of 𝐺 . The eccentric resistance distance sum (ERDS) of 𝐺 is defined as 𝜉 𝑅 ⁡ ( 𝐺 ) = ∑ { 𝑥 , 𝑦 } ⊆ 𝑉 𝐺 ( ɛ 𝐺 ⁡ ( 𝑥 ) + ɛ 𝐺 ⁡ ( 𝑦 ) ) ⁢ 𝑟 𝐺 ⁡ ( 𝑥 , 𝑦 ) , where ɛ 𝐺 ⁡ ( 𝑥 ) is the eccentricity of the vertex 𝑥 . This article focuses on determining extremal graphs with respect to ERDS among certain graph classes. In particular, we characterize the graphs that attain the minimum ERDS within two specific classes, namely, graphs with a fixed number of blocks and graphs with a given number of cut edges. In addition, we identify the unique extremal graph that maximizes the ERDS among all graphs with a prescribed clique number. We obtain these results by studying some edge-grafting transformations on 𝐺 .