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Register a new userThe inertia set of a signed graph
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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 $Sigmasubseteq E$. By $S(G,Sigma)$ we denote the set of all symmetric $Vtimes V$ 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 stable inertia set of a signed graph $(G,Sigma)$ is the set of all pairs $(p,q)$ for which there exists a matrix $Ain S(G,Sigma)$ with $p$ positive and $q$ negative eigenvalues which has the Strong Arnold Property. In this paper, we study the stable inertia set of (signed) graphs.
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A signed graph is a pair $(G,Sigma)$, where $G=(V,E)$ is a graph (in which parallel edges and loops are permitted) with $V={1,ldots,n}$ and $Sigmasubseteq E$. The edges in $Sigma$ are called odd edges and the other edges of $E$ even. By $S(G,Sigma)$ we denote the set of all symmetric $ntimes n$ real matrices $A=[a_{i,j}]$ such that if $a_{i,j} < 0$, then there must be an even edge connecting $i$ and $j$; if $a_{i,j} > 0$, then there must be an odd edge connecting $i$ and $j$; and if $a_{i,j} = 0$, then either there must be an odd edge and an even edge connecting $i$ and $j$, or there are no edges connecting $i$ and $j$. (Here we allow $i=j$.) For a symmetric real matrix $A$, the partial inertia of $A$ is the pair $(p,q)$, where $p$ and $q$ are the number of positive and negative eigenvalues of $A$, respectively. If $(G,Sigma)$ is a signed graph, we define the emph{inertia set} of $(G,Sigma)$ as the set of the partial inertias of all matrices $A in S(G,Sigma)$. In this paper, we present a formula that allows us to obtain the minimal elements of the inertia set of $(G,Sigma)$ in case $(G,Sigma)$ has a $1$-separation using the inertia sets of certain signed graphs associated to the $1$-separation.
In 1982, Zaslavsky introduced the concept of a proper vertex colouring of a signed graph $G$ as a mapping $phicolon V(G)to mathbb{Z}$ such that for any two adjacent vertices $u$ and $v$ the colour $phi(u)$ is different from the colour $sigma(uv)phi(v)$, where is $sigma(uv)$ is the sign of the edge $uv$. The substantial part of Zaslavskys research concentrated on polynomial invariants related to signed graph colourings rather than on the behaviour of colourings of individual signed graphs. We continue the study of signed graph colourings by proposing the definition of a chromatic number for signed graphs which provides a natural extension of the chromatic number of an unsigned graph. We establish the basic properties of this invariant, provide bounds in terms of the chromatic number of the underlying unsigned graph, investigate the chromatic number of signed planar graphs, and prove an extension of the celebrated Brooks theorem to signed graphs.
A signed graph $Gamma(G)$ is a graph with a sign attached to each of its edges, where $G$ is the underlying graph of $Gamma(G)$. The energy of a signed graph $Gamma(G)$ is the sum of the absolute values of the eigenvalues of the adjacency matrix $A(Gamma(G))$ of $Gamma(G)$. The random signed graph model $mathcal{G}_n(p, q)$ is defined as follows: Let $p, q ge 0$ be fixed, $0 le p+q le 1$. Given a set of $n$ vertices, between each pair of distinct vertices there is either a positive edge with probability $p$ or a negative edge with probability $q$, or else there is no edge with probability $1-(p+ q)$. The edges between different pairs of vertices are chosen independently. In this paper, we obtain an exact estimate of energy for almost all signed graphs. Furthermore, we establish lower and upper bounds to the energy of random multipartite signed graphs.
We obtain new bounds for the Laplacian spectral radius of a signed graph. Most of these new bounds have a dependence on edge sign, unlike previously known bounds, which only depend on the underlying structure of the graph. We then use some of these bounds to obtain new bounds for the Laplacian and signless Laplacian spectral radius of an unsigned graph by signing the edges all positive and all negative, respectively.
A mixed graph $widetilde{G}$ is obtained by orienting some edges of $G$, where $G$ is the underlying graph of $widetilde{G}$. The positive inertia index, denoted by $p^{+}(G)$, and the negative inertia index, denoted by $n^{-}(G)$, of a mixed graph $widetilde{G}$ are the integers specifying the numbers of positive and negative eigenvalues of the Hermitian adjacent matrix of $widetilde{G}$, respectively. In this paper, we study the positive and negative inertia index of the mixed unicyclic graph. Moreover, we give the upper and lower bounds of the positive and negative inertia index of the mixed graph, and characterize the mixed graphs which attain the upper and lower bounds respectively.
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