engleski

On the difference of Mostar index and irregularity of graphs

For a connected graph the irregularity irr (G) are  G, the Mostar index Mo(G) and  defined as Mo(G) = uv∈E(G) |n u − n v | and irr (G) = uv∈E(G) |d u − d v |, respec- tively, where d u is the degree of the vertex u of G and n u 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 M(G) = Mo(G) − irr (G) of graphs G. For trees T of order n, we characterize the minimum and second minimum M(T ) of T and the minimum M(T r (T )) of the triangulation graphs T r (T ). The parity of M of cactus graphs is also reported. The effect on M is studied for two local operations of subdivision and contraction of an edge in a tree. A formula for M(S(T )) of the subdivision trees S(T ) and the upper and lower bounds on M(S(T )) − M(T ) are determined with the corresponding extremal trees T . Moreover, three related open problems are proposed to M of graphs.

Mostar index · Irregularity · Tree · Cactus graph · Edge subdivision · Edge contraction

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