On the Estrada index of unicyclic and bicyclic signed graphs

Let $\Gamma=(G, \sigma)$ be a signed graph of order $n$ with eigenvalues $\mu_1,\mu_2,\ldots,\mu_n.$ We define the Estrada index of a signed graph $\Gamma$ as $EE(\Gamma)=\sum_{i=1}^ne^{\mu_i}$. We characterize the signed unicyclic graphs with the maximum Estrada index. The signed graph $\Gamma$ is said to have the pairing property if $\mu$ is an eigenvalue whenever $-\mu$ is an eigenvalue of $\Gamma$ and both $\mu$ and $-\mu$ have the same multiplicities. If $\Gamma_{p}^-(n, m)$ denotes the set of all unbalanced graphs on $n$ vertices and $m$ edges with the pairing property, we determine the signed graphs having the maximum Estrada index in $\Gamma_{p}^-(n, m)$, when $m=n$ and $m=n+1$. Finally, we find the signed graphs among all unbalanced complete bipartite signed graphs having the maximum Estrada index.Comment: 16 pages, 2 figure

Six signed Petersen graphs, and their automorphisms

Up to switching isomorphism there are six ways to put signs on the edges of the Petersen graph. We prove this by computing switching invariants, especially frustration indices and frustration numbers, switching automorphism groups, chromatic numbers, and numbers of proper 1-colorations, thereby illustrating some of the ideas and methods of signed graph theory. We also calculate automorphism groups and clusterability indices, which are not invariant under switching. In the process we develop new properties of signed graphs, especially of their switching automorphism groups.Comment: 39 pp., 7 fi

Spectral Theory for Networks with Attractive and Repulsive Interactions

There is a wealth of applied problems that can be posed as a dynamical system defined on a network with both attractive and repulsive interactions. Some examples include: understanding synchronization properties of nonlinear oscillator;, the behavior of groups, or cliques, in social networks; the study of optimal convergence for consensus algorithm; and many other examples. Frequently the problems involve computing the index of a matrix, i.e. the number of positive and negative eigenvalues, and the dimension of the kernel. In this paper we consider one of the most common examples, where the matrix takes the form of a signed graph Laplacian. We show that the there are topological constraints on the index of the Laplacian matrix related to the dimension of a certain homology group. In certain situations, when the homology group is trivial, the index of the operator is rigid and is determined only by the topology of the network and is independent of the strengths of the interactions. In general these constraints give upper and lower bounds on the number of positive and negative eigenvalues, with the dimension of the homology group counting the number of eigenvalue crossings. The homology group also gives a natural decomposition of the dynamics into "fixed" degrees of freedom, whose index does not depend on the edge-weights, and an orthogonal set of "free" degrees of freedom, whose index changes as the edge weights change. We also present some numerical studies of this problem for large random matrices.Comment: 27 pages; 9 Figure