The hyper-Wiener index is a graph invariant that is used as a structure descriptor for predicting physicochemical properties of organic compounds. We determine the n-vertex unicyclic graphs with the third smallest and the third largest hyper-Wiener indices for $n\ge 5$.

A graph $G$ with no isolated vertex is total restrained domination vertex critical if for any vertex $v$ of $G$ that is not adjacent to a vertex of degree one, the total restrained domination number of $G–v$ is less than the total restrained domination number of $G$. We call these graphs ${\gamma }_{tr}$-vertex critical. If such a graph $G$ has total restrained domination number $k$, we call it $k$-${\gamma }_{tr}$-vertex critical. In this paper, we study matching properties in $4$-${\gamma }_{tr}$-vertex critical graphs of minimum degree at least two.

A generalized weighted digraph $G=(V,E)$ is a digraph with $n$ vertices and $m$ arcs without loops and multiarcs, where each arc is assigned a weight that is a non-negative and symmetric matrix of order $p$. In this paper, we give a sharp upper bound for the spectral radius of generalized weighted digraphs (see Theorem 2.7), which generalizes some other results on the spectral radius of weighted digraphs in [4], [11], and [16].

It has been shown by Bennett et al. in 1998 that a holey Schröder design with $n$ holes of size 2 and one hole of size $u$, i.e., of type $2^{n}u$, exists if $1\le u\le 4$ and $n\ge u+1$ with the exception of $(n,u)\in \{(2,1),(3,1),(3,2)\}$, or $u\ge 16$ and $n\ge \lceil \frac{5u}{4}\rceil +14$. In this paper, we extend this result by showing that, for $1\le u\le 16$, a holey Schröder design of type $2^{n}u$ exists if and only if $n\ge u+1$, with the exception of $(n,u)\in \{(2,1),(3,1),(3,2)\}$ and with the possible exception of $(n,u)\in \{(7,5),(7,6),(11,9),(11,10)\}$. For general $u$, we prove that there exists an HSD($2^{n}u$) for all $u\ge 17$ and $n\ge \lceil \frac{5u}{4}\rceil +4$. Moreover, if $u\ge 35$, then an HSD($2^{n}u$) exists for all $n\ge \lceil \frac{5u}{4}\rceil +1$; if $u\ge 95$, then an HSD($2^{n}u$) exists for all $n\ge \lceil \frac{5u}{4}\rceil –2$. We also improve a well-known result on the existence of holey Schröder designs of type $h^n$ by removing the remaining possible exception of type $64$.

A vertex of a graph is said to be total domination critical if its deletion decreases the total domination number. A graph is said to be total domination vertex critical if all of its vertices, except the supporting vertices, are total domination vertex critical. We show that if $G$ is a connected total domination vertex critical graph with total domination number $k\ge 4$, then the diameter of $G$ is at most $\lfloor \frac{5k-7}{3}\rfloor$.

By computer-assisted approaches and inductive arguments, two curious sums of triple multiplication of binomial coefficients are established in the present paper. The two curious sums arise in proving Melham’s conjecture on odd power sums of Fibonacci numbers, which was confirmed by Xie, Yang and the present author. However, being different from their’s technical way, the method used in the paper is more elementary.

Let $G$ be a graph and $u$ be a vertex of $G$. The transmission index of $u$ in $G$, denoted by $T_G(u)$, is the sum of distances from $u$ to all the other vertices in graph $G$, i.e., $T(u)=T_G(u)=\sum _{v\in V}{d}_G(u,v)$. The Co-PI index [1] is defined as $Co\text{-}PI(G)=\sum _{uv\in E(G)}|T(u)–T(v)|$. In this paper, we give some upper bounds for the Co-PI indices of the join, composition, disjunction, symmetric difference, and corona graph $G_1\circ G_2$.

The purpose of this note is the study of the hypergroups associated with binary relations. New types of matrices, called $i$-very good and regular reversible matrices, are introduced in order to give some properties of the Rosenberg hypergroups related to them. A program written in MATLAB computes the number of these hypergroups up to isomorphism.

Let $A_n$ be the alternating group of degree $n$ with $n>4$. Set $T=\{(123),(132),(12)(3i)\mid 4\le i\le n\}$. The alternating group network, denoted by $A{N}_n$, is defined as the Cayley graph on $A_n$ with respect to $T$. Some properties of $A{N}_n$ have been investigated in [App. Math.—JCU, Ser. A 14 (1998) 235-239; IEEE Trans. Comput. 55 (2006) 1645-1648; Inform. Process. Lett. 110 (2010) 403-409; J. Supercomput. 54 (2010) 206-228]. In this paper, it is shown that the full automorphism group of $A{N}_n$ is the semi-direct product $R(A_n)⋊\text{Aut}(A_n,T)$, where $R(A_n)$ is the right regular representation of $A_n$ and $\text{Aut}(A_n,T)=\{\alpha \in \text{Aut}(A_n)\mid T^{\alpha }=T\}\cong S_{n-3}\times S_2$.

The harmonic index of a graph $G$ is defined as the sum of weights $\frac{2}{\mathrm{deg}(v)+\mathrm{deg}(u)}$ of all edges $uv$ in $E(G)$, where $\mathrm{deg}(v)$ denotes the degree of a vertex $v$ in $V(G)$. In this note, we generalize results of [L. Zhong, The harmonic index on graphs, Appl. Math. Lett. 25 (2012), 561-566] and establish some upper and lower bounds on the harmonic index of $G$.