Solution to a conjecture on the maximal energy of bipartite bicyclic graphs

The energy of a simple graph $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let $C_n$ denote the cycle of order $n$ and $P^{6,6}_n$ the graph obtained from joining two cycles $C_6$ by a path $P_{n-12}$ with its two leaves. Let $\mathscr{B}_n$ denote the class of all bipartite bicyclic graphs but not the graph $R_{a,b}$, which is obtained from joining two cycles $C_a$ and $C_b$ ($a, b\geq 10$ and $a \equiv b\equiv 2\, (\,\textmd{mod}\, 4)$) by an edge. In [I. Gutman, D. Vidovi\'{c}, Quest for molecular graphs with maximal energy: a computer experiment, {\it J. Chem. Inf. Sci.} {\bf41}(2001), 1002--1005], Gutman and Vidovi\'{c} conjectured that the bicyclic graph with maximal energy is $P^{6,6}_n$, for $n=14$ and $n\geq 16$. In [X. Li, J. Zhang, On bicyclic graphs with maximal energy, {\it Linear Algebra Appl.} {\bf427}(2007), 87--98], Li and Zhang showed that the conjecture is true for graphs in the class $\mathscr{B}_n$. However, they could not determine which of the two graphs $R_{a,b}$ and $P^{6,6}_n$ has the maximal value of energy. In [B. Furtula, S. Radenkovi\'{c}, I. Gutman, Bicyclic molecular graphs with the greatest energy, {\it J. Serb. Chem. Soc.} {\bf73(4)}(2008), 431--433], numerical computations up to $a+b=50$ were reported, supporting the conjecture. So, it is still necessary to have a mathematical proof to this conjecture. This paper is to show that the energy of $P^{6,6}_n$ is larger than that of $R_{a,b}$, which proves the conjecture for bipartite bicyclic graphs. For non-bipartite bicyclic graphs, the conjecture is still open.Comment: 9 page

Extremal Structure on Revised Edge-Szeged Index with Respect to Tricyclic Graphs

For a given graph G, Sze*(G)=∑e=uv∈E(G)mu(e)+m0(e)2mv(e)+m0(e)2 is the revised edge-Szeged index of G, where mu(e) and mv(e) are the number of edges of G lying closer to vertex u than to vertex v and the number of edges of G lying closer to vertex v than to vertex u, respectively, and m0(e) is the number of edges equidistant to u and v. In this paper, we identify the lower bound of the revised edge-Szeged index among all tricyclic graphs and also characterize the extremal structure of graphs that attain the bound

Note on the Reformulated Zagreb Indices of Two Classes of Graphs

The reformulated Zagreb indices of a graph are obtained from the original Zagreb indices by replacing vertex degrees with edge degrees, where the degree of an edge is taken as the sum of degrees of its two end vertices minus 2. In this paper, we obtain two upper bounds of the first reformulated Zagreb index among all graphs with p pendant vertices and all graphs having key vertices for which they will become trees after deleting their one key vertex. Moreover, the corresponding extremal graphs which attained these bounds are characterized