Transactions on Combinatorics
Volume:9 Issue: 2, Jun 2020

Let $H$ be a digraph possibly with loops and $D$ a digraph without loops whose arcs are colored with the vertices of $H$ ($D$ is said to be an $H$-colored digraph)‎. ‎A directed walk $W$ in $D$ is said to be an $H$-walk if and only if the consecutive colors encountered on $W$ form a directed walk in $H$‎. ‎A subset $N$ of the vertices of $D$ is said to be an $H$-kernel by walks if (1) for every pair of different vertices in $N$ there is no $H$-walk between them ($N$ is $H$-independent by walks) and (2) for each vertex $u$ in $V$($D$)-$N$ there exists an $H$-walk from $u$ to $N$ in $D$ ($N$ is $H$-absorbent by walks)‎. ‎Suppose that $D$ is a digraph possibly infinite‎. ‎In this paper we will work with the subdivision digraph $S_H$($D$) of $D$‎, ‎where $S_H$($D$) is an $H$-colored digraph defined as follows‎: ‎$V$($S_H$($D$)) = $V$($D$) $cup$ $A$($D$) and $A$($S_H$($D$)) = {($u$,$a$)‎ : ‎$a$ = ($u$,$v$) $in$ $A$($D$)} $cup$ {($a$,$v$)‎ : ‎$a$ = ($u$,$v$) $in$ $A$($D$)}‎, ‎where ($u$‎, ‎$a$‎, ‎$v$) is an $H$-walk in $S_H$($D$) for every $a$ = ($u$,$v$) in $A$($D$)‎. ‎We will show sufficient conditions on $D$ and on $S_H$($D$) which guarantee the existence or uniqueness of $H$-kernels by walks in $S_H$($D$)‎.

The zero forcing number $Z(G)$ of a graph $G$ is the minimum cardinality of a set $S$ with colored (black) vertices which forces the set $V(G)$ to be colored (black) after some times. ``color change rule'': a white vertex is changed to a black vertex when it is the only white neighbor of a black vertex. In this case, we say that the black vertex forces the white vertex. We investigate here the concept of connected zero forcing set and connected zero forcing number. We discusses this subject for special graphs and some products of graphs. Also we introduce the connected propagation time. Graphs with extreme minimum connected propagation times and maximum propagation times $|G|-1$ and $|G|-2$ are characterized.

In this paper‎, ‎we evaluate determinants of some families of Toeplitz--Hessenberg matrices having tribonacci number entries‎. ‎These determinant formulas may also be expressed equivalently as identities that involve sums of products of multinomial coefficients and tribonacci numbers‎. ‎In particular‎, ‎we establish a connection between the tribonacci and the Fibonacci and Padovan sequences via Toeplitz--Hessenberg determinants‎. ‎We then obtain‎, ‎by combinatorial arguments‎, ‎extensions of our determinant formulas in terms of generalized tribonacci sequences satisfying a recurrence of the form $T_n^{(r)}=T_{n-1}^{(r)}+T_{n-2}^{(r)}+T_{n-r}^{(r)}$ for $n geq r$‎, ‎with the appropriate initial conditions‎, ‎where $r geq 3$ is arbitrary‎.

We prove that the graph obtained from the non-nilpotent graph of a finite group by deleting the isolated vertices is connected with diameter at most 3‎. ‎This bound is the best possible‎.

‎Let $x$ be a vertex of a connected graph $G$ and $W subset V(G)$ such that $xnotin W$‎. ‎Then $W$ is called an $x$-Steiner set of textit{G} if $W cup lbrace x rbrace$ is a Steiner set of textit{G}‎. ‎The minimum cardinality of an $x$-textit{Steiner set} of textit{G} is defined as $x$-textit{Steiner number} of textit{G} and denoted by $s_x(G)$‎. ‎Some general properties satisfied by these concepts are studied‎. ‎The $x$-textit{Steiner numbers} of certain classes of graphs are determined‎. ‎Connected graphs of order textit{p} with $x$-Steiner number 1 or $p-1$ are characterized‎. ‎It is shown that for every pair textit{a}‎, ‎textit{b} of integers with $2 leq a leq b$‎, ‎there exists a connected graph textit{G} such that $s(G)} = a$ and $s_{x}(G)=b$ for some vertex $x$ in textit{G}‎, ‎where $s(G)$ is the textit{Steiner number} of a graph‎.