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The revised Szeged index of a graph is defined as S z * ( G ) = ∑ e = u v ∈ E ( n u ( e ) + n 0 ( e ) 2 ) ( n v ( e ) + n 0 ( e ) 2 ) , where n u ( e ) and n v ( e ) are, respectively, the number of vertices of G lying closer to vertex u than to vertex v and the number of vertices of G lying closer to vertex v than to vertex u , and n 0 ( e ) is the number of vertices equidistant to u and v . In the paper, we identify the lower bound of revised Szeged index among all bicyclic graphs, and also characterize the extremal graphs that attain the lower bound. [ABSTRACT FROM AUTHOR]

Mili c ˘ evi c ´ et al. first introduced the reformulated Zagreb indices, which is a generalization of classical Zagreb indices of chemical graph theory. As we know, the Zagreb indices have been found applications in QSPR and QSAR studies. In the paper, we characterize the extremal properties of the first reformulated Zagreb index. We show some graph operations which increase or decrease this index. Furthermore, we determine the sharp bound of the first reformulated Zagreb index among all the extremal tricyclic graphs by these graph operations. [ABSTRACT FROM AUTHOR]

For a given graph G , its edge Szeged index is denoted by S z e (G) = ∑ e = u v ∈ E (G) m u (e) m v (e) , where m u (e) and m v (e) are the number of edges in G with distance to u less than to v and the number of edges in G further (distance) to u than v , respectively. In the paper, the bounds of edge Szeged index on bicyclic graphs are determined. Furthermore, the graphs that achieve the bounds are completely characterized. [ABSTRACT FROM AUTHOR]

The energy of a simple graph is defined as the sum of the absolute values of all the eigenvalues of its adjacency matrix. Jianping found the structure of trees which have maximal energy among all trees of diameter 4 whose centers are of degree t . This paper generalizes the result. We determine the structure of the general tree of diameter 4 and maximal energy. [ABSTRACT FROM AUTHOR]