870 research outputs found

    Saturation numbers in tripartite graphs

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    Given graphs HH and FF, a subgraph GβŠ†HG\subseteq H is an FF-saturated subgraph of HH if F⊈GF\nsubseteq G, but FβŠ†G+eF\subseteq G+e for all e∈E(H)βˆ–E(G)e\in E(H)\setminus E(G). The saturation number of FF in HH, denoted sat(H,F)\text{sat}(H,F), is the minimum number of edges in an FF-saturated subgraph of HH. In this paper we study saturation numbers of tripartite graphs in tripartite graphs. For β„“β‰₯1\ell\ge 1 and n1n_1, n2n_2, and n3n_3 sufficiently large, we determine sat(Kn1,n2,n3,Kβ„“,β„“,β„“)\text{sat}(K_{n_1,n_2,n_3},K_{\ell,\ell,\ell}) and sat(Kn1,n2,n3,Kβ„“,β„“,β„“βˆ’1)\text{sat}(K_{n_1,n_2,n_3},K_{\ell,\ell,\ell-1}) exactly and sat(Kn1,n2,n3,Kβ„“,β„“,β„“βˆ’2)\text{sat}(K_{n_1,n_2,n_3},K_{\ell,\ell,\ell-2}) within an additive constant. We also include general constructions of Kβ„“,m,pK_{\ell,m,p}-saturated subgraphs of Kn1,n2,n3K_{n_1,n_2,n_3} with few edges for β„“β‰₯mβ‰₯p>0\ell\ge m\ge p>0.Comment: 18 pages, 6 figure

    The inertia of weighted unicyclic graphs

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    Let GwG_w be a weighted graph. The \textit{inertia} of GwG_w is the triple In(Gw)=(i+(Gw),iβˆ’(Gw),In(G_w)=\big(i_+(G_w),i_-(G_w), i0(Gw)) i_0(G_w)\big), where i+(Gw),iβˆ’(Gw),i0(Gw)i_+(G_w),i_-(G_w),i_0(G_w) are the number of the positive, negative and zero eigenvalues of the adjacency matrix A(Gw)A(G_w) of GwG_w including their multiplicities, respectively. i+(Gw)i_+(G_w), iβˆ’(Gw)i_-(G_w) is called the \textit{positive, negative index of inertia} of GwG_w, respectively. In this paper we present a lower bound for the positive, negative index of weighted unicyclic graphs of order nn with fixed girth and characterize all weighted unicyclic graphs attaining this lower bound. Moreover, we characterize the weighted unicyclic graphs of order nn with two positive, two negative and at least nβˆ’6n-6 zero eigenvalues, respectively.Comment: 23 pages, 8figure
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