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

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

Regular graphs with maximal energy per vertex

We study the energy per vertex in regular graphs. For every k, we give an upper bound for the energy per vertex of a k-regular graph, and show that a graph attains the upper bound if and only if it is the disjoint union of incidence graphs of projective planes of order k-1 or, in case k=2, the disjoint union of triangles and hexagons. For every k, we also construct k-regular subgraphs of incidence graphs of projective planes for which the energy per vertex is close to the upper bound. In this way, we show that this upper bound is asymptotically tight

Beyond graph energy: norms of graphs and matrices

In 1978 Gutman introduced the energy of a graph as the sum of the absolute values of graph eigenvalues, and ever since then graph energy has been intensively studied. Since graph energy is the trace norm of the adjacency matrix, matrix norms provide a natural background for its study. Thus, this paper surveys research on matrix norms that aims to expand and advance the study of graph energy. The focus is exclusively on the Ky Fan and the Schatten norms, both generalizing and enriching the trace norm. As it turns out, the study of extremal properties of these norms leads to numerous analytic problems with deep roots in combinatorics. The survey brings to the fore the exceptional role of Hadamard matrices, conference matrices, and conference graphs in matrix norms. In addition, a vast new matrix class is studied, a relaxation of symmetric Hadamard matrices. The survey presents solutions to just a fraction of a larger body of similar problems bonding analysis to combinatorics. Thus, open problems and questions are raised to outline topics for further investigation.Comment: 54 pages. V2 fixes many typos, and gives some new materia