Saturation numbers in tripartite graphs

Given graphs $H$ and $F$, a subgraph $G\subseteq H$ is an $F$-saturated subgraph of $H$ if $F\nsubseteq G$, but $F\subseteq G+e$ for all $e\in E(H)\setminus E(G)$. The saturation number of $F$ in $H$, denoted $\text{sat}(H,F)$, is the minimum number of edges in an $F$-saturated subgraph of $H$. In this paper we study saturation numbers of tripartite graphs in tripartite graphs. For $\ell\ge 1$ and $n_1$, $n_2$, and $n_3$ sufficiently large, we determine $\text{sat}(K_{n_1,n_2,n_3},K_{\ell,\ell,\ell})$ and $\text{sat}(K_{n_1,n_2,n_3},K_{\ell,\ell,\ell-1})$ exactly and $\text{sat}(K_{n_1,n_2,n_3},K_{\ell,\ell,\ell-2})$ within an additive constant. We also include general constructions of $K_{\ell,m,p}$-saturated subgraphs of $K_{n_1,n_2,n_3}$ with few edges for $\ell\ge m\ge p>0$.Comment: 18 pages, 6 figure

The inertia of weighted unicyclic graphs

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