In this paper we study the signed Roman dominationnumber of the join of graphs. Specially, we determine it for thejoin of cycles, wheels, fans and friendship graphs.

Suppose $X$ is a simple graph. The $X-$join $Gamma$ of a set ofcomplete or empty graphs ${X_x }_{x in V(X)}$ is a simple graph with the following vertex and edge sets:begin{eqnarray*}V(Gamma) &=& {(x,y) | x in V(X) & y inV(X_x) },\ E(Gamma) &=& {(x,y)(x^prime,y^prime) | xx^prime in E(X) or else x = x^prime & yy^prime in E(X_x)}.end{eqnarray*}The $X-$join graph $Gamma$ is said to be re...

The energy of signed graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. Two signed graphs are said to be equienergetic if they have same energy. In the literature the construction of equienergetic signed graphs are reported. In this paper we obtain the characteristic polynomial and energy of the join of two signed graphs and thereby we give another construction ...

a multicone graph is defined to be join of a clique and a regular graph. a graph $ g $ is  cospectral with  graph $ h $ if their adjacency matrices have the same eigenvalues. a graph $ g $ is  said to be  determined by its spectrum or ds for short, if for any graph $ h $ with $ spec(g)=spec(h)$, we conclude that $ g $ is isomorphic to $ h $. in this paper, we present new classes of  multicone g...

‎    The Narumi-Katayama index is the first topological index defined by the product of some graph theoretical quantities. Let G be a simple graph. Narumi-Katayama index of G is defined as the product of the degrees of the vertices of G. In this paper, we define the Narumi-Katayama polynomial of G. Next, we investigate some properties of this polynomial for graphs and then, we obtain ...

‎Let $f$ be a proper $k$-coloring of a connected graph $G$ and‎ ‎$Pi=(V_1,V_2,ldots,V_k)$ be an ordered partition of $V(G)$ into‎ ‎the resulting color classes‎. ‎For a vertex $v$ of $G$‎, ‎the color‎ ‎code of $v$ with respect to $Pi$ is defined to be the ordered‎ ‎$k$-tuple $c_{{}_Pi}(v)=(d(v,V_1),d(v,V_2),ldots,d(v,V_k))$‎, ‎where $d(v,V_i)=min{d(v,x):~xin V_i}‎, ‎1leq ileq k$‎. ‎If‎ ‎distinct...

The main aim of this study is to characterize new classes of multicone graphs which are determined by their signless Laplacian spectra. A multicone graph is defined to be the join of a clique and a regular graph. Let C and K w denote the Clebsch graph and a complete graph on w vertices, respectively. In this paper, we show that the multicone graphs K w ▽C are determined by their signless ...

Let $G$ be a simple graph of order $n$ and let $k$ an integer such that $1\leq k\leq n-1$. The $k$-token $G^{\{k\}}$ is the whose vertices are $k$-subsets $V(G)$, where two adjacent in whenever their symmetric difference pair $G$. In this paper we study Hamiltonicity graphs some join graphs. As consequence, provide infinite family (containing Hamiltonian non-Hamiltonian graphs) for which Hamilt...