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Register a new userSome signed graphs whose eigenvalues are main
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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.
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A signed graph is a pair $(G,Sigma)$, where $G=(V,E)$ is a graph (in which parallel edges are permitted, but loops are not) with $V={1,...,n}$ and $Sigmasubseteq E$. The edges in $Sigma$ are called odd and the other edges even. By $S(G,Sigma)$ we denote the set of all symmetric $ntimes n$ matrices $A=[a_{i,j}]$ with $a_{i,j}<0$ if $i$ and $j$ are connected by only even edges, $a_{i,j}>0$ if $i$ and $j$ are connected by only odd edges, $a_{i,j}in mathbb{R}$ if $i$ and $j$ are connected by both even and odd edges, $a_{i,j}=0$ if $i ot=j$ and $i$ and $j$ are non-adjacent, and $a_{i,i} in mathbb{R}$ for all vertices $i$. The parameter $ u(G,Sigma)$ of a signed graph $(G,Sigma)$ is the largest nullity of any positive semidefinite matrix $Ain S(G,Sigma)$ that has the Strong Arnold Property. By $K_3^=$ we denote the signed graph obtained from $(K_3,emptyset)$ by adding to each even edge an odd edge in parallel. In this paper, we prove that a signed graph $(G,Sigma)$ has $ u(G,Sigma)leq 2$ if and only if $(G,Sigma)$ has no minor isomorphic to $(K_4,E(K_4))$ or $K_3^=$.
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.
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.
In this paper, we study the characteristic polynomials of the line graphs of generalized Bethe trees. We give an infinite family of such graphs sharing the same smallest eigenvalue. Our family generalizes the family of coronas of complete graphs discovered by Cvetkovic and Stevanovic.
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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