On the Difference of Mostar Index and Irregularity of Graphs

For a connected graph G, the Mostar index Mo(G) and the irregularity irr(G) are defined as Mo(G) = Sigma(uv is an element of E(G)) vertical bar n(u) - n(v)vertical bar and irr(G) = Sigma(uv is an element of E)(G) vertical bar d(u) - d(v) vertical bar, respectively, where d(u) is the degree of the vertex u of G and nu denotes the number of vertices of G which are closer to u than to v for an edge uv. In this paper, we focus on the difference Delta M(G) = Mo(G) - irr(G) of graphs G. For trees T of order n, we characterize the minimum and second minimum Delta M(T) of T and the minimum Delta M(Tr(T)) of the triangulation graphs Tr(T). The parity of Delta M of cactus graphs is also reported. The effect on Delta M is studied for two local operations of subdivision and contraction of an edge in a tree. A formula for Delta M(S(T)) of the subdivision trees S(T) and the upper and lower bounds on Delta M(S(T)) - Delta M(T) are determined with the corresponding extremal trees T. Moreover, three related open problems are proposed to Delta M of graphs.