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Register a new userTwo characterizations of the grid graphs
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In this paper we give two characterizations of the $p times q$-grid graphs as co-edge-regular graphs with four distinct eigenvalues.
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It is not hard to find many complete bipartite graphs which are not determined by their spectra. We show that the graph obtained by deleting an edge from a complete bipartite graph is determined by its spectrum. We provide some graphs, each of which is obtained from a complete bipartite graph by adding a vertex and an edge incident on the new vertex and an original vertex, which are not determined by their spectra.
Characterizing graphs by their spectra is an important topic in spectral graph theory, which has attracted a lot of attention of researchers in recent years. It is generally very hard and challenging to show a given graph to be determined by its spectrum. In Wang~[J. Combin. Theory, Ser. B, 122 (2017):438-451], the author gave a simple arithmetic condition for a family of graphs being determined by their generalized spectra. However, the method applies only to a family of the so called emph{controllable graphs}; it fails when the graphs are non-controllable. In this paper, we introduce a class of non-controllable graphs, called emph{almost controllable graphs}, and prove that, for any pair of almost controllable graphs $G$ and $H$ that are generalized cospectral, there exist exactly two rational orthogonal matrices $Q$ with constant row sums such that $Q^{rm T}A(G)Q=A(H)$, where $A(G)$ and $A(H)$ are the adjacency matrices of $G$ and $H$, respectively. The main ingredient of the proof is a use of the Binet-Cauchy formula. As an application, we obtain a simple criterion for an almost controllable graph $G$ to be determined by its generalized spectrum, which in some sense extends the corresponding result for controllable graphs.
The size-Ramsey number of a graph $F$ is the smallest number of edges in a graph $G$ with the Ramsey property for $F$, that is, with the property that any 2-colouring of the edges of $G$ contains a monochromatic copy of $F$. We prove that the size-Ramsey number of the grid graph on $ntimes n$ vertices is bounded from above by $n^{3+o(1)}$.
A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$, $x eq y$, alternate in $w$ if and only if $(x,y)in E$. Halld{o}rsson et al. have shown that a graph is word-representable if and only if it admits a so-called semi-transitive orientation. A corollary to this result is that any 3-colorable graph is word-representable. Akrobotu et al. have shown that a triangulation of a grid graph is word-representable if and only if it is 3-colorable. This result does not hold for triangulations of grid-covered cylinder graphs, namely, there are such word-representable graphs with chromatic number 4. In this paper we show that word-representability of triangulations of grid-covered cylinder graphs with three sectors (resp., more than three sectors) is characterized by avoiding a certain set of six minimal induced subgraphs (resp., wheel graphs $W_5$ and $W_7$).
Let $G=( V(G), E(G) )$ be a connected graph with vertex set $V(G)$ and edge set $E(G)$. We say a subset $D$ of $V(G)$ dominates $G$ if every vertex in $V setminus D$ is adjacent to a vertex in $D$. A generalization of this concept is $(t,r)$ broadcast domination. We designate certain vertices to be towers of signal strength $t$, which send out signal to neighboring vertices with signal strength decaying linearly as the signal traverses the edges of the graph. We let $mathbb{T}$ be the set of all towers, and we define the signal received by a vertex $vin V(G)$ from a tower $w in mathbb T$ to be $f(v)=sum_{win mathbb{T}}max(0,t-d(v,w))$. Blessing, Insko, Johnson, Mauretour (2014) defined a $(t,r)$ broadcast dominating set, or a $(t,r) $ broadcast, on $G$ as a set $mathbb{T} subseteq V(G) $ such that $f(v)geq r$ for all $vin V(G)$. The minimal cardinality of a $(t, r)$ broadcast on $G$ is called the $(t, r)$ broadcast domination number of $G$. In this paper, we present our research on the $(t,r)$ broadcast domination number for certain graphs including paths, grid graphs, the slant lattice, and the kings lattice.
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