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Register a new userOn automorphisms of the double cover of a circulant graph
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A graph $X$ is said to be unstable if the direct product $X times K_2$ (also called the canonical double cover of $X$) has automorphisms that do not come from automorphisms of its factors $X$ and $K_2$. It is nontrivially unstable if it is unstable, connected, and nonbipartite, and no two distinct vertices of X have exactly the same neighbors. We find three new conditions that each imply a circulant graph is unstable. (These yield infinite families of nontrivially unstable circulant graphs that were not previously known.) We also find all of the nontrivially unstable circulant graphs of order $2p$, where $p$ is any prime number. Our results imply that there does not exist a nontrivially unstable circulant graph of order $n$ if and only if either $n$ is odd, or $n < 8$, or $n = 2p$, for some prime number $p$ that is congruent to $3$ modulo $4$.
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A graph $X$ is said to be unstable if the direct product $X times K_2$ (also called the canonical double cover of $X$) has automorphisms that do not come from automorphisms of its factors $X$ and $K_2$. It is nontrivially unstable if it is unstable, connected, and non-bipartite, and no two distinct vertices of X have exactly the same neighbors. We find all of the nontrivially unstable circulant graphs of valency at most $7$. (They come in several infinite families.) We also show that the instability of each of these graphs is explained by theorems of Steve Wilson. This is best possible, because there is a nontrivially unstable circulant graph of valency $8$ that does not satisfy the hypotheses of any of Wilsons four instability theorems for circulant graphs.
For a graph $G,$ we consider $D subset V(G)$ to be a porous exponential dominating set if $1le sum_{d in D}$ $left( frac{1}{2} right)^{text{dist}(d,v) -1}$ for every $v in V(G),$ where dist$(d,v)$ denotes the length of the smallest $dv$ path. Similarly, $D subset V(G)$ is a non-porous exponential dominating set is $1le sum_{d in D} left( frac{1}{2} right)^{overline{text{dist}}(d,v) -1}$ for every $v in V(G),$ where $overline{text{dist}}(d,v)$ represents the length of the shortest $dv$ path with no internal vertices in $D.$ The porous and non-porous exponential dominating number of $G,$ denoted $gamma_e^*(G)$ and $gamma_e(G),$ are the minimum cardinality of a porous and non-porous exponential dominating set, respectively. The consecutive circulant graph, $C_{n, [ell]},$ is the set of $n$ vertices such that vertex $v$ is adjacent to $v pm i mod n$ for each $i in [ell].$ In this paper we show $gamma_e(C_{n, [ell]}) = gamma_e^*(C_{n, [ell]}) = leftlceil tfrac{n}{3ell +1} rightrceil.$
For a simple graph $G=(V,E),$ let $mathcal{S}_+(G)$ denote the set of real positive semidefinite matrices $A=(a_{ij})$ such that $a_{ij} eq 0$ if ${i,j}in E$ and $a_{ij}=0$ if ${i,j} otin E$. The maximum positive semidefinite nullity of $G$, denoted $operatorname{M}_+(G),$ is $max{operatorname{null}(A)|Ain mathcal{S}_+(G)}.$ A tree cover of $G$ is a collection of vertex-disjoint simple trees occurring as induced subgraphs of $G$ that cover all the vertices of $G$. The tree cover number of $G$, denoted $T(G)$, is the cardinality of a minimum tree cover. It is known that the tree cover number of a graph and the maximum positive semidefinite nullity of a graph are equal for outerplanar graphs, and it was conjectured in 2011 that $T(G)leq M_+(G)$ for all graphs [Barioli et al., Minimum semidefinite rank of outerplanar graphs and the tree cover number, $ Elec. J. Lin. Alg.,$ 2011]. We show that the conjecture is true for certain graph families. Furthermore, we prove bounds on $T(G)$ to show that if $G$ is a connected outerplanar graph on $ngeq 2$ vertices, then $operatorname{M}_+(G)=T(G)leq leftlceilfrac{n}{2}rightrceil$, and if $G$ is a connected outerplanar graph on $ngeq 6$ vertices with no three or four cycle, then $operatorname{M}_+(G)=T(G)leq frac{n}{3}$. We also characterize connected outerplanar graphs with $operatorname{M}_+(G)=T(G)=leftlceilfrac{n}{2}rightrceil.$
A graph is said to be a cover graph if it is the underlying graph of the Hasse diagram of a finite partially ordered set. The direct product G X H of graphs G and H is the graph having vertex set V(G) X V(H) and edge set E(G X H) = {(g_i,h_s)(g_j,h_t): g_ig_j belongs to E(G) and h_sh_t belongs to E(H)}. We prove that the direct product M_m(G) X M_n(H) of the generalized Mycielskians of G and H is a cover graph if and only if G or H is bipartite.
We study the automorphisms of a graph product of finitely-generated abelian groups W. More precisely, we study a natural subgroup Aut* W of Aut W, with Aut* W = Aut W whenever vertex groups are finite and in a number of other cases. We prove a number of structure results, including a semi-direct product decomposition of Aut* W in which one of the factors is Inn W. We also give a number of applications, some of which are geometric in nature.
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