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View article: The Minimum Spectral Radius of $tP_3$- or $K_5$-Saturated Graphs via the Number of $2$-Walks
The Minimum Spectral Radius of $tP_3$- or $K_5$-Saturated Graphs via the Number of $2$-Walks Open
For a given graph $H$, a graph $G$ is $H$-saturated if $G$ does not contain $H$ as a subgraph, but for $e \in E(\overline{G})$, $G+e$ contains $H$ as a subgraph; the spectral saturation number of $H$, written $sat_{\rho}(n,H)$, is the mini…
View article: Eigenvalues and factors: a survey
Eigenvalues and factors: a survey Open
A factor of a graph is a spanning subgraph satisfying some given conditions. An earlier survey of factors can be traced back to the Akiyama and Kano [J. Graph Theory, 1985, 9: 1-42] in which they described the characterization of factors i…
View article: Colorings in digraphs from the spectral radius
Colorings in digraphs from the spectral radius Open
In this paper, we prove that for a digraph D, we have ρ(D)≤maxv∈V(D)∑u∈N−(v)d+(u), where for a vertex v∈V(D), d+(v) is the number of vertices u such that vu is an arc. As a result, we prove χ(D)≤1+maxv∈V(D)∑u∈N−(v)d+(u). We also prove th…
View article: Eigenvalues and [a,b]‐factors in regular graphs
Eigenvalues and [a,b]‐factors in regular graphs Open
For positive integers, , , and , Bollobás, Saito, and Wormald proved some sufficient conditions for an ‐edge‐connected ‐regular graph to have a ‐factor in 1985. Lu gave an upper bound for the third largest eigenvalue in a connected ‐regula…
View article: Eigenvalues and parity factors in graphs
Eigenvalues and parity factors in graphs Open
Let $G$ be a graph and let $g, f$ be nonnegative integer-valued functions defined on $V(G)$ such that $g(v) \le f(v)$ and $g(v) \equiv f(v) \pmod{2}$ for all $v \in V(G)$. A $(g,f)$-parity factor of $G$ is a spanning subgraph $H$ such that…
View article: A Cvetković-type Theorem for coloring of digraphs
A Cvetković-type Theorem for coloring of digraphs Open
In 1972, Cvetković proved that if G is an n-vertex simple graph with the chromatic number k, then its spectral radius is at most the spectral radius of the n-vertex balanced complete k-partite graph. In this paper, we analyze the character…
View article: $K_{r+1}$-saturated graphs with small spectral radius
$K_{r+1}$-saturated graphs with small spectral radius Open
For a graph $H$, a graph $G$ is $H$-saturated if $G$ does not contain $H$ as a subgraph but for any $e \in E(\overline{G})$, $G+e$ contains $H$. In this note, we prove a sharp lower bound for the number of paths and walks on length $2$ in …
View article: Sharp upper bounds on the $k$-independence number in graphs with given minimum and maximum degree
Sharp upper bounds on the $k$-independence number in graphs with given minimum and maximum degree Open
The $k$-independence number of a graph $G$ is the maximum size of a set of vertices at pairwise distance greater than $k$. In this paper, for each positive integer $k$, we prove sharp upper bounds for the $k$-independence number in an $n$-…
View article: Sharp spectral bounds for the edge-connectivity of a regular graph
Sharp spectral bounds for the edge-connectivity of a regular graph Open
Let $λ_2(G)$ and $κ'(G)$ be the second largest eigenvalue and the edge-connectivity of a graph $G$, respectively. Let $d$ be a positive integer at least 3. For $t=1$ or 2, Cioaba proved sharp upper bounds for $λ_2(G)$ in a $d$-regular simp…
View article: Sharp conditions for the existence of an even $[a,b]$-factor in a graph
Sharp conditions for the existence of an even $[a,b]$-factor in a graph Open
Let $a$ and $b$ be positive integers. An even $[a,b]$-factor of a graph $G$ is a spanning subgraph $H$ such that for every vertex $v \in V(G)$, $d_H(v)$ is even and $a \le d_H(v) \le b$. Matsuda conjectured that if $G$ is an $n$-vertex 2-e…
View article: Spectral Bounds for the Connectivity of Regular Graphs with Given Order
Spectral Bounds for the Connectivity of Regular Graphs with Given Order Open
The second-largest eigenvalue and second-smallest Laplacian eigenvalue of a graph are measures of its connectivity. These eigenvalues can be used to analyze the robustness, resilience, and synchronizability of networks, and are related to …
View article: Extremal problems on saturation for the family of $k$-edge-connected graphs
Extremal problems on saturation for the family of $k$-edge-connected graphs Open
Let $\mathcal{F}$ be a family of graphs. A graph $G$ is $\mathcal{F}$-saturated if $G$ contains no member of $\mathcal{F}$ as a subgraph but $G+e$ contains some member of $\mathcal{F}$ whenever $e\in E(\overline{G})$. The saturation number…
View article: The saturation number, spectral radius, and family of $k$-edge-connected graphs
The saturation number, spectral radius, and family of $k$-edge-connected graphs Open
Let $\mathcal{F}$ be a family of graphs. A graph $G$ is $\mathcal{F}$-saturated if $G$ contains no member of $\mathcal{F}$ as a subgraph but $G+e$ contains some member of $\mathcal{F}$ whenever $e\in E(\overline{G})$. The saturation number…
View article: Sharp bounds for the Randic index of graphs with given minimum and maximum degree
Sharp bounds for the Randic index of graphs with given minimum and maximum degree Open
The Randi{\' c} index of a graph $G$, written $R(G)$, is the sum of $\frac 1{\sqrt{d(u)d(v)}}$ over all edges $uv$ in $E(G)$. %let $R(G)=\sum_{uv \in E(G)} \frac 1{\sqrt{d(u)d(v)}}$, which is called the Randi{\' c} index of it. Let $d$ and…
View article: Spectral Bounds for the Connectivity of Regular Graphs with Given Order
Spectral Bounds for the Connectivity of Regular Graphs with Given Order Open
The second-largest eigenvalue and second-smallest Laplacian eigenvalue of a graph are measures of its connectivity. These eigenvalues can be used to analyze the robustness, resilience, and synchronizability of networks, and are related to …
View article: On the Wiener index, distance cospectrality and transmission regular graphs
On the Wiener index, distance cospectrality and transmission regular graphs Open
In this paper, we investigate various algebraic and graph theoretic properties of the distance matrix of a graph. Two graphs are $D$-cospectral if their distance matrices have the same spectrum. We construct infinite pairs of $D$-cospectra…
View article: The second largest eigenvalue and vertex-connectivity of regular multigraphs
The second largest eigenvalue and vertex-connectivity of regular multigraphs Open
Let $μ_2(G)$ be the second smallest Laplacian eigenvalue of a graph $G$. The vertex connectivity of $G$, written $κ(G)$, is the minimum size of a vertex set $S$ such that $G-S$ is disconnected. Fiedler proved that $μ_2(G) \le κ(G)$ for a n…
View article: Algebraic connectivity of multigraphs
Algebraic connectivity of multigraphs Open
Let $\mu_2(G)$ be the second smallest Laplacian eigenvalue of a graph $G$, and let $\kappa(G)$ be the minimum size of a vertex set $S$ such that $G-S$ is disconnected. Fiedler proved that $\mu_2(G) \le \kappa(G)$ for a non-complete simple …