On the $A_alpha$ spectral radius and $A_alpha$ energy of non-strongly connected digraphs

Let $A_alpha(G)$ be the $A_alpha$-matrix of a digraph $G$ and $lambda_{alpha 1}, lambda_{alpha 2}, ldots, lambda_{alpha n}$ be the eigenvalues of $A_alpha(G)$. Let $rho_alpha(G)$ be the $A_alpha$ spectral radius of $G$ and $E_alpha(G)=sum_{i=1}^n lambda_{alpha i}^2$ be the $A_alpha$ energy of $G$ by using second spectral moment. Let $mathcal{G}_n^m$ be the set of non-strongly connected digraphs with order $n$, which contain a unique strong component with order $m$ and some directed trees which are hung on each vertex of the strong component. In this paper, we characterize the digraph which has the maximal $A_alpha$ spectral radius and the maximal (minimal) $A_alpha$ energy in $mathcal{G}_n^m$.

On the $A_alpha$ spectral radius of strongly connected digraphs

Let $G$ be a digraph with adjacency matrix $A(G)$. Let $D(G)$ be the diagonal matrix with outdegrees of vertices of $G$. Nikiforov cite{Niki} proposed to study the convex combinations of the adjacency matrix and diagonal matrix of the degrees of undirected graphs. Liu et al. cite{LWCL} extended the definition to digraphs. For any real $alphain[0,1]$, the matrix $A_alpha(G)$ of a digraph $G$ is defined as $$A_alpha(G)=alpha D(G)+(1-alpha)A(G).$$ The largest modulus of the eigenvalues of $A_alpha(G)$ is called the $A_alpha$ spectral radius of $G$, denoted by $lambda_alpha(G)$. This paper proves some extremal results about the spectral radius $lambda_alpha(G)$ that generalize previous results about $lambda_0(G)$ and $lambda_{frac{1}{2}}(G)$. In particular, we characterize the extremal digraph with the maximum (or minimum) $A_alpha$ spectral radius among all $widetilde{infty}$-digraphs and $widetilde{theta}$-digraphs on $n$ vertices. Furthermore, we determine the digraphs with the second and the third minimum $A_alpha$ spectral radius among all strongly connected bicyclic digraphs. For $0leqalphaleqfrac{1}{2}$, we also determine the digraphs with the second, the third and the fourth minimum $A_alpha$ spectral radius among all strongly connected digraphs on $n$ vertices. Finally, we characterize the digraph with the minimum $A_alpha$ spectral radius among all strongly connected bipartite digraphs which contain a complete bipartite subdigraph.

Integer colorings with no rainbow 3-term arithmetic progression

In this paper, we study the rainbow ErdH{o}s-Rothschild problem with respect to 3-term arithmetic progressions. We obtain the asymptotic number of $r$-colorings of $[n]$ without rainbow 3-term arithmetic progressions, and we show that the typical colorings with this property are 2-colorings. We also prove that $[n]$ attains the maximum number of rainbow 3-term arithmetic progression-free $r$-colorings among all subsets of $[n]$. Moreover, the exact number of rainbow 3-term arithmetic progression-free $r$-colorings of $mathbb{Z}_p$ is obtained, where $p$ is any prime and $mathbb{Z}_p$ is the cyclic group of order $p$.

The ErdH{o}s-Gy{a}rf{a}s function with respect to Gallai-colorings

For fixed $p$ and $q$, an edge-coloring of the complete graph $K_n$ is said to be a $(p, q)$-coloring if every $K_p$ receives at least $q$ distinct colors. The function $f(n, p, q)$ is the minimum number of colors needed for $K_n$ to have a $(p, q)$-coloring. This function was introduced about 45 years ago, but was studied systematically by ErdH{o}s and Gy{a}rf{a}s in 1997, and is now known as the ErdH{o}s-Gy{a}rf{a}s function. In this paper, we study $f(n, p, q)$ with respect to Gallai-colorings, where a Gallai-coloring is an edge-coloring of $K_n$ without rainbow triangles. Combining the two concepts, we consider the function $g(n, p, q)$ that is the minimum number of colors needed for a Gallai-$(p, q)$-coloring of $K_n$. Using the anti-Ramsey number for $K_3$, we have that $g(n, p, q)$ is nontrivial only for $2leq qleq p-1$. We give a general lower bound for this function and we study how this function falls off from being equal to $n-1$ when $q=p-1$ and $pgeq 4$ to being $Theta(log n)$ when $q = 2$. In particular, for appropriate $p$ and $n$, we prove that $g=n-c$ when $q=p-c$ and $cin {1,2}$, $g$ is at most a fractional power of $n$ when $q=lfloorsqrt{p-1}rfloor$, and $g$ is logarithmic in $n$ when $2leq qleq lfloorlog_2 (p-1)rfloor+1$.

The generalized Tur{a}n number of spanning linear forests

Let $mathcal{F}$ be a family of graphs. A graph $G$ is called textit{$mathcal{F}$-free} if for any $Fin mathcal{F}$, there is no subgraph of $G$ isomorphic to $F$. Given a graph $T$ and a family of graphs $mathcal{F}$, the generalized Tur{a}n number of $mathcal{F}$ is the maximum number of copies of $T$ in an $mathcal{F}$-free graph on $n$ vertices, denoted by $ex(n,T,mathcal{F})$. A linear forest is a graph whose connected components are all paths or isolated vertices. Let $mathcal{L}_{n,k}$ be the family of all linear forests of order $n$ with $k$ edges and $K^*_{s,t}$ a graph obtained from $K_{s,t}$ by substituting the part of size $s$ with a clique of the same size. In this paper, we determine the exact values of $ex(n,K_s,mathcal{L}_{n,k})$ and $ex(n,K^*_{s,t},mathcal{L}_{n,k})$. Also, we study the case of this problem when the textit{host graph} is bipartite. Denote by $ex_{bip}(n,T,mathcal{F})$ the maximum possible number of copies of $T$ in an $mathcal{F}$-free bipartite graph with each part of size $n$. We determine the exact value of $ex_{bip}(n,K_{s,t},mathcal{L}_{n,k})$. Our proof is mainly based on the shifting method.

The bounds of the spectral radius of general hypergraphs in terms of clique number

The spectral radius (or the signless Laplacian spectral radius) of a general hypergraph is the maximum modulus of the eigenvalues of its adjacency (or its signless Laplacian) tensor. In this paper, we firstly obtain a lower bound of the spectral radius (or the signless Laplacian spectral radius) of general hypergraphs in terms of clique number. Moreover, we present a relation between a homogeneous polynomial and the clique number of general hypergraphs. As an application, we finally obtain an upper bound of the spectral radius of general hypergraphs in terms of clique number.

Extremal problems and results related to Gallai-colorings

A Gallai-coloring (Gallai-$k$-coloring) is an edge-coloring (with colors from ${1, 2, ldots, k}$) of a complete graph without rainbow triangles. Given a graph $H$ and a positive integer $k$, the $k$-colored Gallai-Ramsey number $GR_k(H)$ is the minimum integer $n$ such that every Gallai-$k$-coloring of the complete graph $K_n$ contains a monochromatic copy of $H$. In this paper, we consider two extremal problems related to Gallai-$k$-colorings. First, we determine upper and lower bounds for the maximum number of edges that are not contained in any rainbow triangle or monochromatic triangle in a $k$-edge-coloring of $K_n$. Second, for $ngeq GR_k(K_3)$, we determine upper and lower bounds for the minimum number of monochromatic triangles in a Gallai-$k$-coloring of $K_{n}$, yielding the exact value for $k=3$. Furthermore, we determine the Gallai-Ramsey number $GR_k(K_4+e)$ for the graph on five vertices consisting of a $K_4$ with a pendant edge.

Iota energy orderings of bicyclic signed digraphs

The concept of energy of a signed digraph is extended to iota energy of a signed digraph. The energy of a signed digraph $S$ is defined by $E(S)=sum_{k=1}^n|text{Re}(z_k)|$, where $text{Re}(z_k)$ is the real part of eigenvalue $z_k$ and $z_k$ is the eigenvalue of the adjacency matrix of $S$ with $n$ vertices, $k=1,2,ldots,n$. Then the iota energy of $S$ is defined by $E(S)=sum_{k=1}^n|text{Im}(z_k)|$, where $text{Im}(z_k)$ is the imaginary part of eigenvalue $z_k$. In this paper, we consider a special graph class for bicyclic signed digraphs $mathcal{S}_n$ with $n$ vertices which have two vertex-disjoint signed directed even cycles. We give two iota energy orderings of bicyclic signed digraphs, one is including two positive or two negative directed even cycles, the other is including one positive and one negative directed even cycles.

On the Capacity of MIMO Optical Wireless Channels

This paper studies the capacity of a general multiple-input multiple-output (MIMO) free-space optical intensity channel under a per-input-antenna peak-power constraint and a total average-power constraint over all input antennas. The focus is on the scenario with more transmit than receive antennas. In this scenario, different input vectors can yield identical distributions at the output, when they result in the same image vector under multiplication by the channel matrix. We first determine the most energy-efficient input vectors that attain each of these image vectors. Based on this, we derive an equivalent capacity expression in terms of the image vector, and establish new lower and upper bounds on the capacity of this channel. The bounds match when the signal-to-noise ratio (SNR) tends to infinity, establishing the high-SNR asymptotic capacity. We also characterize the low-SNR slope of the capacity of this channel.

The generalized distance matrix of digraphs

Let $D(G)$ and $D^Q(G)= Diag(Tr) + D(G)$ be the distance matrix and distance signless Laplacian matrix of a simple strongly connected digraph $G$, respectively, where $Diag(Tr)=textrm{diag}(D_1,D_2,$ $ldots,D_n)$ be the diagonal matrix with vertex transmissions of the digraph $G$. To track the gradual change of $D(G)$ into $D^Q(G)$, in this paper, we propose to study the convex combinations of $D(G)$ and $Diag(Tr)$ defined by $$D_alpha(G)=alpha Diag(Tr)+(1-alpha)D(G), 0leq alphaleq1.$$ This study reduces to merging the distance spectral and distance signless Laplacian spectral theories. The eigenvalue with the largest modulus of $D_alpha(G)$ is called the $D_alpha$ spectral radius of $G$, denoted by $mu_alpha(G)$. We determine the digraph which attains the maximum (or minimum) $D_alpha$ spectral radius among all strongly connected digraphs. Moreover, we also determine the digraphs which attain the minimum $D_alpha$ spectral radius among all strongly connected digraphs with given parameters such as dichromatic number, vertex connectivity or arc connectivity.