Complete solution to a problem on the maximal energy of unicyclic bipartite 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. Denote by $C_n$ the cycle, and $P_n^{6}$ the unicyclic graph obtained by connecting a vertex of $C_6$ with a leaf of $P_{n-6}$\,. Caporossi et al. conjecture that the unicyclic graph with maximal energy is $P_n^6$ for $n=8,12,14$ and $n\geq 16$. In``Y. Hou, I. Gutman and C. Woo, Unicyclic graphs with maximal energy, {\it Linear Algebra Appl.} {\bf 356}(2002), 27--36", the authors proved that $E(P_n^6)$ is maximal within the class of the unicyclic bipartite $n$-vertex graphs differing from $C_n$\,. And they also claimed that the energy of $C_n$ and $P_n^6$ is quasi-order incomparable and left this as an open problem. In this paper, by utilizing the Coulson integral formula and some knowledge of real analysis, especially by employing certain combinatorial techniques, we show that the energy of $P_n^6$ is greater than that of $C_n$ for $n=8,12,14$ and $n\geq 16$, which completely solves this open problem and partially solves the above conjecture.Comment: 8 page

On a conjecture about tricyclic graphs with maximal energy

For a given simple graph $G$, the energy of $G$, denoted by $\mathcal {E}(G)$, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix, which was defined by I. Gutman. The problem on determining the maximal energy tends to be complicated for a given class of graphs. There are many approaches on the maximal energy of trees, unicyclic graphs and bicyclic graphs, respectively. Let $P^{6,6,6}_n$ denote the graph with $n\geq 20$ vertices obtained from three copies of $C_6$ and a path $P_{n-18}$ by adding a single edge between each of two copies of $C_6$ to one endpoint of the path and a single edge from the third $C_6$ to the other endpoint of the $P_{n-18}$. Very recently, Aouchiche et al. [M. Aouchiche, G. Caporossi, P. Hansen, Open problems on graph eigenvalues studied with AutoGraphiX, {\it Europ. J. Comput. Optim.} {\bf 1}(2013), 181--199] put forward the following conjecture: Let $G$ be a tricyclic graphs on $n$ vertices with $n=20$ or $n\geq22$, then $\mathcal{E}(G)\leq \mathcal{E}(P_{n}^{6,6,6})$ with equality if and only if $G\cong P_{n}^{6,6,6}$. Let $G(n;a,b,k)$ denote the set of all connected bipartite tricyclic graphs on $n$ vertices with three vertex-disjoint cycles $C_{a}$, $C_{b}$ and $C_{k}$, where $n\geq 20$. In this paper, we try to prove that the conjecture is true for graphs in the class $G\in G(n;a,b,k)$, but as a consequence we can only show that this is true for most of the graphs in the class except for 9 families of such graphs.Comment: 32 pages, 12 figure

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

On the maximal energy tree with two maximum degree vertices

For a simple graph $G$, the energy $E(G)$ is defined as the sum of the absolute values of all eigenvalues of its adjacent matrix. For $\Delta\geq 3$ and $t\geq 3$, denote by $T_a(\Delta,t)$ (or simply $T_a$) the tree formed from a path $P_t$ on $t$ vertices by attaching $\Delta-1$ $P_2$'s on each end of the path $P_t$, and $T_b(\Delta, t)$ (or simply $T_b$) the tree formed from $P_{t+2}$ by attaching $\Delta-1$ $P_2$'s on an end of the $P_{t+2}$ and $\Delta -2$ $P_2$'s on the vertex next to the end. In [X. Li, X. Yao, J. Zhang and I. Gutman, Maximum energy trees with two maximum degree vertices, J. Math. Chem. 45(2009), 962--973], Li et al. proved that among trees of order $n$ with two vertices of maximum degree $\Delta$, the maximal energy tree is either the graph $T_a$ or the graph $T_b$, where $t=n+4-4\Delta\geq 3$. However, they could not determine which one of $T_a$ and $T_b$ is the maximal energy tree. This is because the quasi-order method is invalid for comparing their energies. In this paper, we use a new method to determine the maximal energy tree. It turns out that things are more complicated. We prove that the maximal energy tree is $T_b$ for $\Delta\geq 7$ and any $t\geq 3$, while the maximal energy tree is $T_a$ for $\Delta=3$ and any $t\geq 3$. Moreover, for $\Delta=4$, the maximal energy tree is $T_a$ for all $t\geq 3$ but $t=4$, for which $T_b$ is the maximal energy tree. For $\Delta=5$, the maximal energy tree is $T_b$ for all $t\geq 3$ but $t$ is odd and $3\leq t\leq 89$, for which $T_a$ is the maximal energy tree. For $\Delta=6$, the maximal energy tree is $T_b$ for all $t\geq 3$ but $t=3,5,7$, for which $T_a$ is the maximal energy tree. One can see that for most $\Delta$, $T_b$ is the maximal energy tree, $\Delta=5$ is a turning point, and $\Delta=3$ and 4 are exceptional cases.Comment: 16 page

The approach to criticality in sandpiles

A popular theory of self-organized criticality relates the critical behavior of driven dissipative systems to that of systems with conservation. In particular, this theory predicts that the stationary density of the abelian sandpile model should be equal to the threshold density of the corresponding fixed-energy sandpile. This "density conjecture" has been proved for the underlying graph Z. We show (by simulation or by proof) that the density conjecture is false when the underlying graph is any of Z^2, the complete graph K_n, the Cayley tree, the ladder graph, the bracelet graph, or the flower graph. Driven dissipative sandpiles continue to evolve even after a constant fraction of the sand has been lost at the sink. These results cast doubt on the validity of using fixed-energy sandpiles to explore the critical behavior of the abelian sandpile model at stationarity.Comment: 30 pages, 8 figures, long version of arXiv:0912.320