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Let $G$ be a strongly connected digraph with $n$ vertices and $m$ arcs. For any real $\alpha\in[0,1]$, the $A_\alpha$ matrix of a digraph $G$ is defined as $$A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G),$$ where $A(G)$ is the adjacency matrix of $G$ and $D(G)$ is the outdegrees diagonal matrix of $G$. The eigenvalue of $A_\alpha(G)$ with the largest modulus is called the $A_\alpha$ spectral radius of $G$, denoted by $\lambda_{\alpha}(G)$. In this paper, we first obtain an upper bound on $\lambda_{\alpha}(G)$ for $\alpha\in[\frac{1}{2},1)$. Employing this upper bound, we prove that for two strongly connected digraphs $G_1$ and $G_2$ with $n\ge4$ vertices and $m$ arcs, and $\alpha\in [\frac{1}{\sqrt{2}},1)$, if the maximum outdegree $\Delta^+(G_1)\ge 2\alpha(1-\alpha)(m-n+1)+2\alpha$ and $\Delta^+(G_1)>\Delta^+(G_2)$, then $\lambda_\alpha(G_1)>\lambda_\alpha(G_2)$. Moreover, We also give another upper bound on $\lambda_{\alpha}(G)$ for $\alpha\in[\frac{1}{2},1)$. Employing this upper bound, we prove that for two strongly connected digraphs with $m$ arcs, and $\alpha\in[\frac{1}{2},1)$, if the maximum outdegree $\Delta^+(G_1)>\frac{2m}{3}+1$ and $\Delta^+(G_1)>\Delta^+(G_2)$, then $\lambda_\alpha(G_1)+\frac{1}{4}>\lambda_\alpha(G_2)$.

Let $Q(G)=D(G)+A(G)$ be the signless Laplacian matrix of a simple graph $G$, where $D(G)$ and $A(G)$ are the degree diagonal matrix and the adjacency matrix of $G$, respectively. The largest eigenvalue of $Q(G)$, denoted by $q(G)$, is called the signless Laplacian spectral radius of $G$. Let $\theta(l_{1},l_{2},l_{3})$ denote the theta graph which consists of two vertices connected by three internally disjoint paths with length $l_{1}$, $l_{2}$ and $l_{3}$. Let $F_{n}$ be the friendship graph consisting of $\frac{n-1}{2}$ triangles which intersect in exactly one common vertex for odd $n\geq3$ and obtained by hanging an edge to the center of $F_{n-1}$ for even $n\geq4$. Let $S_{n,k}$ denote the graph obtained by joining each vertex of $K_{k}$ to $n-k$ isolated vertices. Let $S_{n,k}^{+}$ denote the graph obtained by adding an edge to the two isolated vertices of $S_{n,k}$. In this paper, firstly, we show that if $G$ is $\theta(1,2,2)$-free, then $q(G)\leq q(F_{n})$, unless $G\cong F_{n}$. Secondly, we show that if $G$ is $\theta(1,2,3)$-free, then $q(G)\leq q(S_{n,2})$, unless $G\cong S_{n,2}$. Finally, we show that if $G$ is $\{\theta(1,2,2),F_{5}\}$-free, then $q(G)\leq q(S_{n,1}^{+})$, unless $G\cong S_{n,1}^{+}$., Comment: 12pages, 3 figures

A graph $G$ is $F$-free if $G$ does not contain $F$ as a subgraph. Let $\mathcal{G}(m, F)$ denote the family of $F$-free graphs with $m$ edges and without isolated vertices. Let $S_{n,k}$ denote the graph obtained by joining every vertex of $K_{k}$ to $n-k$ isolated vertices and $S_{n,k}^{t}$ denote the graph obtained from $S_{n-t,k}$ by attaching $t$ pendant vertices to the maximal degree vertex of $S_{n-t,k}$, respectively. Denote by $H_{n}$ the fan graph obtain from $n-1$-vertex path plus a vertex adjacent to each vertex of the path. Particularly, the graph $H_{5}$ is also known as the gem. Zhang and Wang [Discrete Math. 347(2024)114171] and Yu, Li and Peng [arXiv: 2404. 03423] showed that every gem-free graph $G$ with $m$ edges satisfies $\rho(G)\leq \rho(S_{\frac{m+3}{2},2})$. In this paper, we show that if $G\in \mathcal{G}(m, H_{5})\setminus S_{\frac{m+3}{2},2}$ be a graph of odd size $m\geq23$, then $\rho(G)\leq \rho(S_{\frac{m+5}{2},2}^{2})$, and equality holds if and only if $G\cong S_{\frac{m+5}{2},2}^{2}$., Comment: 10pages. arXiv admin note: text overlap with arXiv:2411.05304

The $Q$-index of graph $G$ is the largest eigenvalue of the signless Laplacian matrix of $G$. Wang [Discrete Appl. Math. 356(2024)] proved the sharp upper bounds on the $Q$-index of leaf-free graphs with given size and characterized the corresponding extremal graphs. A graph is $2$ leaves-free if it has no two pendent vertices. In this paper, we give sharp upper bounds on the $Q$-index of 2 leaves-free graphs with given size and characterize the corresponding extremal graphs., Comment: 21 pages, 5 figures

A graph $G$ is $F$-free if $G$ does not contain $F$ as a subgraph. Let $\rho(G)$ be the spectral radius of a graph $G$. Let $\theta(1,p,q)$ denote the theta graph, which is obtained by connecting two distinct vertices with three internally disjoint paths with lengths $1, p, q$, where $p\leq q$. Let $S_{n,k}$ denote the graph obtained by joining every vertex of $K_{k}$ to $n-k$ isolated vertices and $S_{n,k}^{-}$ denote the graph obtained from $S_{n,k}$ by deleting an edge incident to a vertex of degree $k$, respectively. In this paper, we show that if $\rho(G)\geq\rho(S_{\frac{m+4}{2},2}^{-})$ for a graph $G$ with even size $m\geq 92$, then $G$ contains a $\theta(1,3,3)$ unless $G\cong S_{\frac{m+4}{2},2}^{-}$., Comment: 14pages, 1 figures. arXiv admin note: text overlap with arXiv:2410.07721 by other authors

Let $T_{4p}=\langle a,b\mid a^{2p}=1,a^p=b^2, b^{-1}ab=a^{-1}\rangle$ be the dicyclic group of order $4p$. A Cayley digraph over $T_{4p}$ is called a dicirculant digraph. In this paper, we calculate the number of (connected) dicirculant digraphs of order $4p$ ($p$ prime) up to isomorphism by using the P\'olya Enumeration Theorem. Moreover, we get the number of (connected) dicirculant digraphs of order $4p$ ($p$ prime) and out-degree $k$ for every $k$.

Let $\mathcal{F}$ be a family of $r$-uniform hypergraphs. Denote by $\ex^{\mathrm{conn}}_r(n,\mathcal{F})$ the maximum number of hyperedges in an $n$-vertex connected $r$-uniform hypergraph which contains no member of $\mathcal{F}$ as a subhypergraph. Denote by $\mathcal{B}C_k$ the Berge cycle of length $k$, and by $\mathcal{B}P_k$ the Berge path of length $k$. F\"{u}redi, Kostochka and Luo, and independently Gy\H{o}ri, Salia and Zamora determined $\ex^{\mathrm{conn}}_r(n,\mathcal{B}P_k)$ provided $k$ is large enough compared to $r$ and $n$ is sufficiently large. For the case $k\le r$, Kostochka and Luo obtained an upper bound for $\ex^{\mathrm{conn}}_r(n,\mathcal{B}P_k)$. In this paper, we continue investigating the case $k\le r$. We precisely determine $\ex^{\mathrm{conn}}_r(n,\mathcal{B}P_k)$ when $n$ is sufficiently large and $n$ is not a multiple of~$r$. For the case $k=r+1$, we determine $\ex^{\mathrm{conn}}_r(n,\mathcal{B}P_k)$ asymptotically.

Let $H_7$ denote the $7$-vertex \textit{fan graph} consisting of a $6$-vertex path plus a vertex adjacent to each vertex of the path. Let $K_3 \vee \frac{m-3}{3}K_1$ be the graph obtained by joining each vertex of a triangle $K_3$ to $\frac{m-3}{3}$ isolated vertices. In this paper, we show that if $G$ is an $H_{7}$-free graph with size $m\geq 33$, then the spectral radius $ \rho(G)\leq 1+\sqrt{m-2},$ equality holds if and only if $G\cong K_3 \vee \frac{m-3}{3}K_1$ (possibly, with some isolated vertices)., Comment: 10 pages

We consider the wiretap channel, where the individual channel uses have memory or are influenced by an adversary. We analyze the explicit and computationally efficient construction of information-theoretically secure coding schemes which use the inverse of an extractor and an error-correcting code. These schemes are known to achieve secrecy capacity on a large class of memoryless wiretap channels. We show that this also holds for certain channel types with memory. In particular, they can achieve secrecy capacity on channels where an adversary can pick a sequence of ``states'' governing the channel's behavior, as long as, given every possible state, the channel is strongly symmetric., Comment: 25 pages

We study the fundamental limits of covert communications over general memoryless additive-noise channels. We assume that the legitimate receiver and the eavesdropper share the same channel and therefore see the same outputs. Under mild integrability assumptions, we find a general upper bound on the square-root scaling constant, which only involves the variance of the logarithm of the probability density function of the noise. Furthermore, we show that, under some additional assumptions, this upper bound is tight. We also provide upper bounds on the length of the secret key required to achieve the optimal scaling., Comment: version to appear in IEEE Transactions on Information Theory

In this paper, we study dual quaternion and dual complex unit gain graphs and their spectral properties in a unified frame of dual unit gain graphs. Unit dual quaternions represent rigid movements in the 3D space, and have wide applications in robotics and computer graphics. Dual complex numbers found application in brain science recently. We establish the interlacing theorem for dual unit gain graphs, and show that the spectral radius of a dual unit gain graph is always not greater than the spectral radius of the underlying graph, and these two radii are equal if and only if the dual gain graph is balanced. By using the dual cosine functions, we establish the closed form of eigenvalues of adjacency and Laplacian matrices of dual complex and quaternion unit gain cycles. We then show the coefficient theorem holds for dual unit gain graphs. Similar results hold for the spectral radius of the Laplacian matrix of the dual unit gain graph too.

An $r$-graph $H$ is a hypergraph consisting of a nonempty set of vertices $V$ and a collection of $r$-element subsets of $V$ we refer to as the edges of $H$. An $r$-graph $H$ is called linear if any two edges of $H$ intersect in at most one vertex. Let $F$ and $H$ be two linear $r$-graphs. If $H$ contains no copy of $F$, then $H$ is called $F$-free. The linear Tur\'{a}n number of $F$, denoted by $ex_r^{lin}(n,F)$, is the maximum number of edges in any $F$-free $n$-vertex linear $r$-graph. The crown $C_{13}$ (or $E_4$) is a linear 3-graph which is obtained from three pairwise disjoint edges by adding one edge that intersects all three of them in one vertex. In 2022, Gy\'{a}rf\'{a}s, Ruszink\'{o} and S\'{a}rk\"{o}zy initiated the study of $ex_3^{lin}(n,F)$ for different choices of an acyclic 3-graph $F$. They determined the linear Tur\'{a}n numbers for all linear 3-graphs with at most 4 edges, except the crown. They established lower and upper bounds for $ex_3^{lin}(n,C_{13})$. In fact, their lower bound on $ex_3^{lin}(n,C_{13})$ is essentially tight, as was shown in a recent paper by Tang, Wu, Zhang and Zheng. In this paper, we generalize the notion of a crown to linear $r$-graphs for $r\ge 3$, and also generalize the above results to linear $r$-graphs.

A graph G is said to be M-integral (resp. A-integral, D-integral, DL-integral or DQ-integral) if all eigenvalues of its matrix M (resp. adjacency matrix A(G), distance matrix D(G), distance Laplacian matrix DL(G) or distance signless Laplacian matrix DQ(G)) are integers. Lu et al. [Discrete Math, 346 (2023)] defined the generalized wheel graph GW(a, m, n) as the join of two regular graphs aKm and Cn, and obtained all D-integral generalized wheel graphs. Based on the above research, in this paper, we determine all DL-integral and DQ-integral generalized wheel graphs respectively. As byproducts, we give a sufficient and necessary condition for the join of two regular graphs G1 and G2 to be DL-integral, from which we can get infinitely many new classes of DL-integral graphs according to the large number of research results about the A-integral graphs.

In this paper, we investigate the existence of pretty good fractional revival on Cayley graphs over dicyclic groups. We first give a necessary and sufficient description for Cayley graphs over dicyclic groups admitting pretty good fractional revival. By this description, we give some sufficient conditions for Cayley graphs over dicyclic groups admitting or not admitting pretty good fractional revival.

The capacity of a state-dependent discrete memoryless channel (SD-DMC) is derived for the setting where a message-cognizant rate-limited helper observes the state sequence noncausally, produces its description, and provides the description to both encoder and decoder.

Let $G=(V(G), E(G))$ be a graph with vertex set $V(G)$ and edge set $E(G)$. A graph is $ID$-factor-critical if for every independent set $I$ of $G$ whose size has the same parity as $|V(G)|$, $G-I$ has a perfect matching. For two positive integers $a$ and $b$ with $a\leq b$, let $h$: $E(G)\rightarrow [0, 1]$ be a function on $E(G)$ satisfying $a\leq\sum _{e\in E_{G}(v_{i})}h(e)\leq b$ for any vertex $v_{i}\in V(G)$. Then the spanning subgraph with edge set $E_{h}$, denoted by $G[E_{h}]$, is called a fractional $[a, b]$-factor of $G$ with indicator function $h$, where $E_{h}=\{e\in E(G)\mid h(e)>0\}$ and $E_{G}(v_{i})=\{e\in E(G)\mid e$ is incident with $v_{i}$ in $G$\}. A graph is defined as a fractional $[a, b]$-deleted graph if for any $e\in E(G)$, $G-e$ contains a fractional $[a, b]$-factor. For any integer $k\geq 1$, a graph has a $k$-factor if it contains a $k$-regular spanning subgraph. In this paper, we firstly give a distance spectral radius condition of $G$ to guarantee that $G$ is $ID$-factor-critical. Furthermore, we provide sufficient conditions in terms of distance spectral radius and distance signless Laplacian spectral radius for a graph to contain a fractional $[a, b]$-factor, fractional $[a, b]$-deleted-factor and $k$-factor., Comment: 10 pages

A formula is derived for the capacity of the Gaussian channel with a benevolent message-cognizant rate-limited helper that provides a noncausal description of the noise to the encoder and decoder. This capacity is strictly larger than when the helper is message oblivious, with the difference being particularly pronounced at low signal-to-noise ratios. It is shown that in this setup, a feedback link from the receiver to the encoder does not increase capacity. However, in the presence of such a link, said capacity can be achieved even if the helper is oblivious to the transmitted message.

Let $C_{m}$ be a cycle with length $m.$ The $k$-uniform hypercycle with length $m$ obtained by adding $k-2$ new vertices in every edge of $C_{m},$ denoted by $C_{m,k}.$ In this paper, we obtain some trace formulas of uniform hypercycles with length four. Moreover, we give the characteristic polynomials of uniform hypercycles with length four.

The $A_{\alpha}$-matrix of a graph $G$ is the convex linear combination of the adjacency matrix $A(G)$ and the diagonal matrix of vertex degrees $D(G)$, i.e., $A_{\alpha}(G) = \alpha D(G) + (1 - \alpha)A(G)$, where $0\leq\alpha \leq1$. The $\alpha$-index of $G$ is the largest eigenvalue of $A_\alpha(G)$. Particularly, the matrix $A_0(G)$ (resp. $2A_{\frac{1}{2}}(G)$) is exactly the adjacency matrix (resp. signless Laplacian matrix) of $G$. He, Li and Feng [arXiv:2301.06008 (2023)] determined the extremal graphs with maximum adjacency spectral radius among all graphs of sufficiently large order without intersecting triangles and quadrangles as a minor, respectively. Motivated by the above results of He, Li and Feng, in this paper we characterize the extremal graphs with maximum $\alpha$-index among all graphs of sufficiently large order without intersecting triangles and quadrangles as a minor for any $0<\alpha<1$, respectively. As by-products, we determine the extremal graphs with maximum signless Laplacian radius among all graphs of sufficiently large order without intersecting triangles and quadrangles as a minor, respectively., Comment: 15 pages

In this paper, we investigate the existence of fractional revival on semi-Cayley graphs over finite abelian groups. We give some necessary and sufficient conditions for semi-Cayley graphs over finite abelian groups admitting fractional revival. We also show that integrality is necessary for some semi-Cayley graphs admitting fractional revival. Moreover, we characterize the minimum time when semi-Cayley graphs admit fractional revival. As applications, we give examples of certain Cayley graphs over the generalized dihedral groups and generalized dicyclic groups admitting fractional revival.

Shimokawa, Han, and Amari proposed a "quantization and binning" scheme for distributed binary hypothesis testing. We propose a simple improvement on the receiver's guessing rule in this scheme. This attains a better exponent of the error probability of the second type.

Let $\mathcal{F}$ be a family of $r$-uniform hypergraphs, and let $H$ be an $r$-uniform hypergraph. Then $H$ is called $\mathcal{F}$-free if it does not contain any member of $\mathcal{F}$ as a subhypergraph. The Tur\'{a}n number of $\mathcal{F}$, denoted by $ex_r(n,\mathcal{F})$, is the maximum number of hyperedges in an $\mathcal{F}$-free $n$-vertex $r$-uniform hypergraph. Our current results are motivated by earlier results on Tur\'{a}n numbers of star forests and hypergraph star forests. In particular, Lidick\'{y}, Liu and Palmer [Electron. J. Combin. 20 (2013)] determined the Tur\'{a}n number $ex(n,F)$ of a star forest $F$ for sufficiently large $n$. Recently, Khormali and Palmer [European. J. Combin. 102 (2022) 103506] generalized the above result to three different well-studied hypergraph settings, but restricted to the case that all stars in the hypergraph star forests are identical. We further generalize these results to general hypergraph star forests., Comment: arXiv admin note: substantial text overlap with arXiv:2001.05631 by other authors

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of vertex degrees of $G$. For any real $\alpha \in [0,1]$, Nikiforov defined the $A_\alpha$-matrix of a graph $G$ as $A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G)$. The largest eigenvalue of $A_\alpha(G)$ is called the $\alpha$-index or the $A_\alpha$-spectral radius of $G$. A graph is minimally $k$-(edge)-connected if it is $k$-(edge)-connected and deleting any arbitrary chosen edge always leaves a graph which is not $k$-(edge)-connected. In this paper, we characterize the minimally 2-edge-connected graphs and minimally 3-connected graph with given order having the maximum $\alpha$-index for $\alpha \in [\frac{1}{2},1)$, respectively., Comment: arXiv admin note: text overlap with arXiv:2301.03389

The $k$-th Laplacian spectral moment of a digraph $G$ is defined as $\sum_{i=1}^n \lambda_i^k$, where $\lambda_i$ are the eigenvalues of the Laplacian matrix of $G$ and $k$ is a nonnegative integer. For $k=2$, this invariant is better known as the Laplacian energy of $G$. We extend recently published results by characterizing the digraphs which attain the minimal and maximal Laplacian energy within classes of digraphs with a fixed dichromatic number. We also determine sharp bounds for the third Laplacian spectral moment within the special subclass which we define as join digraphs. We leave the full characterization of the extremal digraphs for $k\ge 3$ as an open problem.

A memoryless state sequence governing the behavior of a memoryless state-dependent channel is to be described causally to an encoder wishing to communicate over said channel. Given the maximal-allowed description rate, we seek the description that maximizes the Shannon capacity. It is shown that the maximum need not be achieved by a memoryless (symbol-by-symbol) description. Such descriptions are, however, optimal when the receiver is cognizant of the state sequence or when the description is allowed to depend on the message. For other cases, a block-Markov scheme with backward decoding is proposed., Comment: To appear in the IEEE Transactions on Information Theory

Let $G=(V(G), E(G))$ be a graph with vertex set $V(G)$ and edge set $E(G)$. For $k\geq2$ and given any subset $S\subseteq|V(G)|$ with $|S|=k$, if a graph $G$ of order $|V(G)|\geq k+1$ always has a spanning tree $T$ such that $S$ is precisely the set of leaves of $T$, then the graph $G$ is a $k$-leaf-connected graph. A graph $G$ is called Hamilton-connected if any two vertices of $G$ are connected by a Hamilton path. Based on the definitions of $k$-leaf-connected and Hamilton-connected, we known that a graph is $2$-leaf-connected if and only if it is Hamilton-connected. During the past decades, there have been many results of sufficient conditions for Hamilton-connected with respect to topological indices. In this paper, we present sufficient conditions for a graph $G$ to be $k$-leaf-connected in terms of the Zagreb index, the reciprocal degree distance or the hyper-Zagreb index. Furthermore, we use the first Zagreb index and hyper-Zagreb index of the complement graph $\overline{G}$ to give sufficient conditions for a graph $G$ to be $k$-leaf-connected., Comment: 19 pages, conference or other essential info

We extend previous results on covert communication over the additive white Gaussian noise channel to two other types of additive noise channels. The first is the Gaussian channel with memory, where the noise sequence is a Gaussian vector with an arbitrary invertible covariance matrix. We show that the fundamental limit for covert communication over such a channel is the same as over the channel with white, i.e., memoryless, Gaussian noise. The second type of channel we consider is one with memoryless generalized Gaussian noise. For such a channel we prove a general upper bound on the dominant term in the maximum number of nats that can be covertly communicated over n channel uses. When the shape parameter p of the generalized Gaussian noise distribution is in the interval (0, 1], we also prove a matching lower bound., Comment: To be presented at ITW 2023

For integer $k\geq2,$ a graph $G$ is called $k$-leaf-connected if $|V(G)|\geq k+1$ and given any subset $S\subseteq V(G)$ with $|S|=k,$ $G$ always has a spanning tree $T$ such that $S$ is precisely the set of leaves of $T.$ Thus a graph is $2$-leaf-connected if and only if it is Hamilton-connected. In this paper, we present a best possible condition based upon the size to guarantee a graph to be $k$-leaf-connected, which not only improves the results of Gurgel and Wakabayashi [On $k$-leaf-connected graphs, J. Combin. Theory Ser. B 41 (1986) 1-16] and Ao, Liu, Yuan and Li [Improved sufficient conditions for $k$-leaf-connected graphs, Discrete Appl. Math. 314 (2022) 17-30], but also extends the result of Xu, Zhai and Wang [An improvement of spectral conditions for Hamilton-connected graphs, Linear Multilinear Algebra, 2021]. Our key approach is showing that an $(n+k-1)$-closed non-$k$-leaf-connected graph must contain a large clique if its size is large enough. As applications, sufficient conditions for a graph to be $k$-leaf-connected in terms of the (signless Laplacian) spectral radius of $G$ or its complement are also presented., Comment: 15 pages, 2 figures

Let $\rho(G)$ be the spectral radius of a graph $G$ with $m$ edges. Let $S_{m-k+1}^{k}$ be the graph obtained from $K_{1,m-k}$ by adding $k$ disjoint edges within its independent set. Nosal's theorem states that if $\rho(G)>\sqrt{m}$, then $G$ contains a triangle. Zhai and Shu showed that any non-bipartite graph $G$ with $m\geq26$ and $\rho(G)\geq\rho(S_{m}^{1})>\sqrt{m-1}$ contains a quadrilateral unless $G\cong S_{m}^{1}$ [M.Q. Zhai, J.L. Shu, Discrete Math. 345 (2022) 112630]. Wang proved that if $\rho(G)\geq\sqrt{m-1}$ for a graph $G$ with size $m\geq27$, then $G$ contains a quadrilateral unless $G$ is one of four exceptional graphs [Z.W. Wang, Discrete Math. 345 (2022) 112973]. In this paper, we show that any non-bipartite graph $G$ with size $m\geq51$ and $\rho(G)\geq\rho(S_{m-1}^{2})>\sqrt{m-2}$ contains a quadrilateral unless $G$ is one of three exceptional graphs. Moreover, we show that if $\rho(G)\geq\rho(S_{\frac{m+4}{2},2}^{-})$ for a graph $G$ with even size $m\geq74$, then $G$ contains a $C_{5}^{+}$ unless $G\cong S_{\frac{m+4}{2},2}^{-}$, where $C_{t}^{+}$ denotes the graph obtained from $C_{t}$ and $C_{3}$ by identifying an edge, $S_{n,k}$ denotes the graph obtained by joining each vertex of $K_{k}$ to $n-k$ isolated vertices and $S_{n,k}^{-}$ denotes the graph obtained by deleting an edge incident to a vertex of degree two, respectively., Comment: 16 pages, 4 figures

For any real $\alpha \in [0,1]$, Nikiforov defined the $A_\alpha$-matrix of a graph $G$ as $A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the diagonal matrix of vertex degrees of $G$, respectively. The largest eigenvalue of $A_\alpha(G)$ is called the $\alpha$-index or the $A_\alpha$-spectral radius of $G$. A graph is minimally $k$-connected if it is $k$-connected and deleting any arbitrary chosen edge always leaves a graph which is not $k$-connected. In this paper, we characterize the extremal graphs with the maximum $\alpha$-index for $\alpha \in [\frac{1}{2},1)$ among all minimally 2-connected graphs with given order or size, respectively., Comment: 15 pages, 1 figure

Let $G$ be a graph with adjacency matrix $A(G)$ and degree diagonal matrix $D (G)$. In 2017, Nikiforov defined the matrix $A_\alpha(G) = \alpha D(G) + (1-\alpha)A(G)$ for any real $\alpha\in[0,1]$. The largest eigenvalue of $A_\alpha(G)$ is called the $A_\alpha$ spectral radius or the $A_\alpha$-index of $G$. Let $\mathcal{G}_{n,k}^d$ be the set of $k$-connected graphs of order $n$ with diameter $d$. In this paper, we determine the graphs with maximum $A_\alpha$ spectral radius among all graphs in $\mathcal{G}_{n,k}^d$ for any $\alpha\in[0,1)$, where $k\geq2$ and $d\geq2$. We generalizes the results about adjacency matrix of Theorem 3.6 in [P. Huang, W.C. Shiu, P.K. Sun, Linear Algebra Appl., 488 (2016) 350--362] and the results about signless Laplacian matrix of Theorem 3.4 in [P. Huang, J.X. Li, W.C. Shiu, Linear Algebra Appl., 617 (2021) 78--99]. Furthermore, we also obtain the upper and lower bounds of the extremal graph in $\mathcal{G}_{n,k}^d$.

Let $G$ be a graph with adjacency matrix $A(G)$ and degree diagonal matrix $D (G)$. In 2017, Nikiforov [Appl. Anal. Discrete Math., 11 (2017) 81--107] defined the matrix $A_\alpha(G) = \alpha D(G) + (1-\alpha)A(G)$ for any real $\alpha\in[0,1]$. The largest eigenvalue of $A(G)$ is called the spectral radius of $G$, while the largest eigenvalue of $A_\alpha(G)$ is called the $A_\alpha$ spectral radius of $G$. Let $\mathcal{G}_{n,i}$ be the set of graphs of order $n$ with independence number $i$. Recently, for all graphs in $\mathcal{G}_{n,i}$ having the minimum or the maximum $A$, $Q$ and $A_\alpha$ spectral radius where $i\in\{1,2,\lfloor\frac{n}{2}\rfloor\,\lceil\frac{n}{2}\rceil+1,n-3,n-2,n-1\}$, there are some results have been given by Xu, Li and Sun et al., respectively. In 2021, Luo and Guo [Discrete Math., 345 (2022) 112778] determined all graphs in $\mathcal{G}_{n,n-4}$ having the minimum spectral radius. In this paper, we characterize the graphs in $\mathcal{G}_{n,n-4}$ having the minimum and the maximum $A_\alpha$ spectral radius for $\alpha\in[\frac{1}{2},1)$, respectively.

In this paper, we investigate the existence of fractional revival on Cayley graphs over finite abelian groups. We give a necessary and sufficient condition for Cayley graphs over finite abelian groups to have fractional revival. As applications, the existence of fractional revival on circulant graphs and cubelike graphs are characterized.

We generalize the notions of Laplacian and signless Laplacian Estrada index to uniform hypergraphs. For an $r$-uniform hypergraph $H,$ we derive an order $r+1$ trace formula of the (signless) Laplacian tensor of $H.$ Among others by using this trace formula, we obtain lower bounds for the signless Laplacian Estrada index and upper bounds for the Laplacian Estrada index. Moreover, we establish a bound involving both the Laplacian Estrada index and Laplacian energy of a uniform hypergraph., Comment: 14 pages, 0 figures

A linear $r$-uniform hypergraph is called acycilc if it can be constructed starting from one single edge then at each step adding a new edge that intersect the union of the vertices of the previous edges in at most one vertex. Recently, Gy\'{a}rf\'{a}s, Ruszink\'{o} and S\'{a}rk\''{o}zy initiated the study of the linear Tur\'{a}n numbers of acyclic linear triple systems. In this paper, we extend their results to linear quadruple systems. Here, we concentrate on small trees, paths and matchings. For the case of small trees, we find that for a linear tree $T$, $ex^{lin}_{4}(n,T)$ relates to difficult problems on Steiner system $S(2,4,n)$ For example, we show that $ex^{lin}_{4}(n, P_4)\le \frac{5n}{4}$ with equality holds if and only if the linear quadruple system is the disjoint union of $S(2,4,16)$. Denote by $E^{+}_{4}$ the linear tree consisting of three pairwise disjoint quadruples and a fourth one intersecting all of them. We prove that $12\lfloor\frac{n-4}{9}\rfloor\le ex^{lin}_{4}(n, E^{+}_4)\le \frac{14(n-s)}{9}$, where $s$ is the number of vertices in $G$ with degree at least 8. Denote by $M_k$ and $P_k$ the set of $k$ pairwise disjoint quadruples and the linear path with $k$ quadruples, respectively. For the case of paths, we show that $ex^{lin}_{4}(n, P_k)\le 2.5kn$. For the case of matchings, we prove that for fixed $k$ and sufficiently large $n$, $ex^{lin}_{4}(n, M_k)=g(n,k)$ where $g(n,k$) denotes the maximum number of quadruples that can intersect $k-1$ vertices in a linear quadruple system on $n$ vertices., Comment: 18 pages, 3 figures