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Register a new userNordhaus-Gaddum type inequality for the fractional matching number of a graph
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The fractional matching number of a graph G, is the maximum size of a fractional matching of G. The following sharp lower bounds for a graph G of order n are proved, and all extremal graphs are characterized in this paper. (1)The sum of the fractional matching number of a graph G and the fractional matching number of its complement is not less than n/2 , where n is not less than 2. (2) If G and its complement are non-empty, then the sum of the fractional matching number of a graph G and the fractional matching number of its complement is not less than (n+1)/2, where n is not less than 28. (3) If G and its complement have no isolated vertices, then the sum of the fractional matching number of a graph G and the fractional matching number of its complement is not less than (n+4)/2, where n is not less than 28.
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Let G be a simple graph. A coloring of vertices of G is called (i) a 2-proper coloring if vertices at distance 2 receive distinct colors; (ii) an injective coloring if vertices possessing a common neighbor receive distinct colors; (iii) a square coloring if vertices at distance at most 2 receive distinct colors. In this paper, we study inequalities of Nordhaus-Guddam type for the 2-proper chromatic number, the injective chromatic number, and the square chromatic number.
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.
Building upon the notion of Gutman index $operatorname{SGut}(G)$, Mao and Das recently introduced the Steiner Gutman index by incorporating Steiner distance for a connected graph $G$. The emph{Steiner Gutman $k$-index} $operatorname{SGut}_k(G)$ of $G$ is defined by $operatorname{SGut}_k(G)$ $=sum_{Ssubseteq V(G), |S|=k}left(prod_{vin S}deg_G(v)right) d_G(S)$, in which $d_G(S)$ is the Steiner distance of $S$ and $deg_G(v)$ is the degree of $v$ in $G$. In this paper, we derive new sharp upper and lower bounds on $operatorname{SGut}_k$, and then investigate the Nordhaus-Gaddum-type results for the parameter $operatorname{SGut}_k$. We obtain sharp upper and lower bounds of $operatorname{SGut}_k(G)+operatorname{SGut}_k(overline{G})$ and $operatorname{SGut}_k(G)cdot operatorname{SGut}_k(overline{G})$ for a connected graph $G$ of order $n$, $m$ edges and maximum degree $Delta$, minimum degree $delta$.
A complex unit gain graph (or ${mathbb T}$-gain graph) is a triple $Phi=(G, {mathbb T}, varphi)$ (or $(G, varphi)$ for short) consisting of a simple graph $G$, as the underlying graph of $(G, varphi)$, the set of unit complex numbers $mathbb{T}= { z in C:|z|=1 }$ and a gain function $varphi: overrightarrow{E} rightarrow mathbb{T}$ with the property that $varphi(e_{i,j})=varphi(e_{j,i})^{-1}$. In this paper, we prove that $2m(G)-2c(G) leq r(G, varphi) leq 2m(G)+c(G)$, where $r(G, varphi)$, $m(G)$ and $c(G)$ are the rank of the Hermitian adjacency matrix $H(G, varphi)$, the matching number and the cyclomatic number of $G$, respectively. Furthermore, the complex unit gain graphs $(G, mathbb{T}, varphi)$ with $r(G, varphi)=2m(G)-2c(G)$ and $r(G, varphi)=2m(G)+c(G)$ are characterized. These results generalize the corresponding known results about undirected graphs, mixed graphs and signed graphs. Moreover, we show that $2m(G-V_{0}) leq r(G, varphi) leq 2m(G)+b(G)$ holds for any subset $V_0$ of $V(G)$ such that $G-V_0$ is acyclic and $b(G)$ is the minimum integer $|S|$ such that $G-S$ is bipartite for $S subset V(G)$.
We review the theory of Cheeger constants for graphs and quantum graphs and their present and envisaged applications.
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