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In this paper, we introduce the concepts of the plain eigenvalue, the main-plain index and the refined spectrum of graphs. We focus on the graphs with two main and two plain eigenvalues and give some characterizations of them.
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In his survey Beyond graph energy: Norms of graphs and matrices (2016), Nikiforov proposed two problems concerning characterizing the graphs that attain equality in a lower bound and in a upper bound for the energy of a graph, respectively. We show that these graphs have at most two nonzero distinct absolute eigenvalues and investigate the proposed problems organizing our study according to the type of spectrum they can have. In most cases all graphs are characterized. Infinite families of graphs are given otherwise. We also show that all graphs satifying the properties required in the problems are integral, except for complete bipartite graphs $K_{p,q}$ and disconnected graphs with a connected component $K_{p,q}$, where $pq$ is not a perfect square.
Let $G$ be a graph. For a subset $X$ of $V(G)$, the switching $sigma$ of $G$ is the signed graph $G^{sigma}$ obtained from $G$ by reversing the signs of all edges between $X$ and $V(G)setminus X$. Let $A(G^{sigma})$ be the adjacency matrix of $G^{sigma}$. An eigenvalue of $A(G^{sigma})$ is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. Let $S_{n,k}$ be the graph obtained from the complete graph $K_{n-r}$ by attaching $r$ pendent edges at some vertex of $K_{n-r}$. In this paper we prove that there exists a switching $sigma$ such that all eigenvalues of $G^{sigma}$ are main when $G$ is a complete multipartite graph, or $G$ is a harmonic tree, or $G$ is $S_{n,k}$. These results partly confirm a conjecture of Akbari et al.
We present the first steps towards the determination of the signed graphs for which the adjacency matrix has all but at most two eigenvalues equal to 1 or -1. Here we deal with the disconnected, the bipartite and the complete signed graphs. In addition, we present many examples which cannot be obtained from an unsigned graph or its negative by switching.
A Neumaier graph is a non-complete edge-regular graph containing a regular clique. In this paper we give some sufficient and necessary conditions for a Neumaier graph to be strongly regular. Further we show that there does not exist Neumaier graphs with exactly four distinct eigenvalues. We also determine the Neumaier graphs with smallest eigenvalue -2.
A theory of orientation on gain graphs (voltage graphs) is developed to generalize the notion of orientation on graphs and signed graphs. Using this orientation scheme, the line graph of a gain graph is studied. For a particular family of gain graphs with complex units, matrix properties are established. As with graphs and signed graphs, there is a relationship between the incidence matrix of a complex unit gain graph and the adjacency matrix of the line graph.
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