let $d$ be a finite and simple digraph with vertex set $v(d)$‎.‎a signed total roman $k$-dominating function (str$k$df) on‎‎$d$ is a function $f:v(d)rightarrow{-1‎, ‎1‎, ‎2}$ satisfying the conditions‎‎that (i) $sum_{xin n^{-}(v)}f(x)ge k$ for each‎‎$vin v(d)$‎, ‎where $n^{-}(v)$ consists of all vertices of $d$ from‎‎which arcs go into $v$‎, ‎and (ii) every vertex $u$ for which‎‎$f(u)=-1$ has a...

2 + )*$ *" $ M 7 " 3 $ 4 5 $ + ,$ ) $ $ ( " , * ! $ !$( $/ H + 6 $ " +$ " $ +$ $( )*$ *"$ ( 6 " + $/ + $ + !$ + $ + ! , $ " ) " " ($ 0$ " )*$ " * + , $ *"$ " $ + 0 / +$ " $*$ "$ $ $ $ *$ " ! " 6 *" , , $ $ $ , !$( " / +$ " , *$ .+ # $ $+ $ *"$ " $( ($( " , + *" ($ * $ + , *$ .+ ( + (+ $ $ $ + $ +/ + )*$ 6 " .$ $* $+ $ , + $0 ( "$ " , *$ " . " $ 1 $* $ + + 6 "$ , 0$ " 1+ "$ $/ 2 + )*$ *" H $ $ $...

7 ! 8+ 90.+ % -+ : + 0 /0 + & . :. 9 03 & 3 03 1 . 0 + 0 0 + + -9 + & 0 0 . 0.+ . 3 09 0 ;. 0 + 03 + + + 3 : 1 . + : <+-. + = : + 2 0 + +9 0 4 3 0 : + 0 + + + 0 > ;. / + + . :?+ 0 1 . 0 + +>0 & + 00 > 0 / 0 + -.: + !! . = /+ 0 0 :?+ 0 & + . . ;. + . ;. + 0 . 0 0 > = . + ;. , + 0 . + 0 + 0 0 , <. ;. 9 3 : 03 + 90.+ : + 0 0 + + 1 . ? +.+ % + ;. , + : . 0 0 + & + 90.+ & 0 0 1 . 9 .+ & 3 / + + & 0 ...

ABSTRACT In a signed graph, each link is labeled with either a positive or a negative sign. This is particularly appropriate to model polarized systems. Such a graph can be characterized through the notion of structural balance, which relies on the partitioning of the graph into internally solidary but mutually hostile subgroups. In this work, we show that signed graphs can be used to model and...

In this paper we prove that no signed graph on the complete graph Kp, p ≥ 6, is graceful, and we also give a characterization of graceful signed graphs on Kp, p ≤ 5. This implies that there is no subset A of cardinality p ≥ 6 from the set {0, 1, . . . , (p 2 ) − n}, n ≤ 1 2 ( p 2 ) , such that each element of the set {1, 2, . . . , n} occurs exactly twice and each element of the remaining set {...

An important property of chordal graphs is that these graphs are characterized by existence of perfect elimination orderings on their vertex sets. In this paper, we generalize the notion of perfect elimination orderings to signed graphs, and give a characterization for graphs admitting such orderings, together with characterizations restricted to some subclasses and further properties of those ...

We study e(λ) = inf spec ( — A + λV) and examine when e(λ) < 0 for all λ + 0. We prove that cλ ^ e(λ) ^ dλ for suitable V and all small \λ\.

! " # $ % & & & ' & ! ' ' & & ! & ' ! & ! '! ! & ! ' ' ' & & # % ! ' ( ) * ! & + ' ) & # , ! ' ' & # ! -& ! ." & ! / ! / & . & & ! ' ' + # , / + & ! ' ' + & )& + 0 & # + ! & & ' " & 1 ! ' + + ) ! & # + & ' ! ' 2 3 2 ' & & 4 ' ! ' # * & & ' + # # * & 5 6 & ' & ! '! # , & # 7 + ! 4 5 8 ' 9 ! & + # # + ' 9 % ' & ' 2 && & ' ' ' # , + ' 4& ' + ' & ' ' # + ' " 5 : & & & ! ! ) ' ! & ' ' # ; & ' & ! ' ...