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A signed graphΣis an ordered pair (G,σ) that consists of a underlying graphG=(V,E)and a sign mapping called signatureσfrom Eto the sign set {+,-}. In this article, we provide another way of looking at the Cartesian product of a path graph and an arbitrary signed graph Σ. We then present the adjacency spectrum and Laplacian spectrum of the Cartesian product in terms of the spectrum and Laplacian spectrum of Σ, respectively. We further provide an upper bound and lower bound for the respective energies. As applications, the results in this article are used (1) to construct a family of infinitely many cospectral and Laplacian cospectral graphs and (2) to compute the adjacency (respectively, Laplacian) spectrum of some known classes of graphs.

For a commutative ring Rwith unity (1≠0), the zero-divisor graph of R, denoted by Γ(R), is a simple graph with vertices as elements of Rand two distinct vertices are adjacent whenever the product of the vertices is zero. Further, its signed zero-divisor graph is an ordered pair ΓΣ(R):=(Γ(R),σ), where for an edge xy, σ(xy)is ‘+’ if either xor yor both is a nonzero zero-divisor and ‘−’ otherwise. This article aims at gaining a deeper insight into signed zero-divisor graphs by investigating properties such as balancing, clusterability, sign-compatibility, consistency, C-sign-compatibility, and C-consistency.

A signed graph(or sigraphin short) is an ordered pair $$S = (S^u, \sigma )$$ S=(Su,σ) , where $$S^u$$ Su is a graph $$G = (V, E)$$ G=(V,E) and $$\sigma :E\rightarrow \{+,-\}$$ σ:E→{+,-} is a function from the edge set $$E$$ E of $$S^u$$ Su into the set $$\{+, -\}$$ {+,-} . The canonical marking on $$S$$ S is defined as: for each vertex $$v \in V(S)$$ v∈V(S) , $$\displaystyle \mu _\sigma (v) = \prod _{e_j \in E_{v}} \sigma (e_j),$$ μσ(v)=∏ej∈Evσ(ej), where $$E_{v}$$ Ev is the set of edges $$e_j$$ ej incident at $$v$$ v in $$S$$ S . A vertex $$v$$ v is called negative if the value of marking of $$v$$ v is negative. Let $$S$$ S is canonically marked, then a cycle $$Z$$ Z in $$S$$ S is said to be canonically consistent if it contains an even number of negative vertices. If every cycle in $$S$$ S is canonically consistent, then $$S$$ S is called canonically consistent. In this paper, we characterize canonically consistent semi-total point sigraphs.

The Cayley sum graph is a graph whose vertex comprises elements of an abelian group Gand edges are the sum of these vertices belonging to a subset of G, namely, S. We introduce Cayley sum signed graph by giving the sign to these edges. An edge receives a positive sign if any of the incident vertices belong to S; otherwise, it receives a negative sign. We discuss the balance, clusterability and some properties of derived Cayley sum signed graph.

The Cayley sum graph is a graph whose vertex comprises elements of an abelian group G, and edges are the sum of these vertices belonging to a subset of G, namely, S. We introduce the Cayley sum signed graph by giving the sign to these edges. An edge receives a positive sign if any of the incident vertices belong to S. Otherwise, it receives a negative sign. We discuss the properties of the Cayley sum signed graph.

Let Rbe a finite commutative ring with identity. The co-maximal graph Γ(R)is a graph with vertices as elements of R, where two distinct vertices aand bare adjacent if and only if Ra+Rb=R. Also, Γ2(R)is the subgraph of Γ(R)induced by non-unit elements and Γ2′(R)=Γ2(R)\J(R)where J(R) is Jacobson radical. In this paper, we characterize the rings for which the graphs Γ2′(R)and L(Γ2′(R))are planar. Also, we characterize rings for which Γ2′(R)and L(Γ2′(R))are split graphs, and obtain nullity of co-maximal graph for local rings and non-local rings along with domination number on co-maximal graph.

The square graph G2of a graph G=(V,E)is a graph with same vertex set as G, and the vertices are adjacent in G2when their distance in Gis at most two. In this paper, we characterize signed graph(or sigraph) which is a square root signed graph of some signed graph. Also, we find whether for a given signed graph its square signed graph and line of square signed graph are balanced. Each theorem is supported by respective algorithms.

Social network analysis has been a subject of interest to many sociologists, computer scientists, psychologists and mathematicians for many centuries now. Lately another aspect of the social network, specific kind of links (friendship/enmity, trust/distrust, like/dislike) has been studied vigorously. In this paper, we try to explore and analyse these networks mathematically by taking a small subnetwork and predicting its properties and behaviour. Let $$G=(V,E)$$ G=(V,E) be a graph then square graph of Gis obtained by adding to Gedges which connect pairs of vertices of Gat a distance two apart. In this paper we discuss few properties of square signed graphs along with their algorithms.

The tensor product$$G \times H$$ G×H of two graphs Gand His a graph such that the vertex set of $$G \times H$$ G×H is the cartesian product $$V(G) \times V(H)$$ V(G)×V(H) and two vertices $$(u_1, u_2)$$ (u1,u2) and $$(v_1, v_2)$$ (v1,v2) are adjacent in $$G \times H$$ G×H if and only if $$u_1$$ u1 is adjacent to $$v_1$$ v1 in Gand $$u_2$$ u2 is adjacent to $$v_2$$ v2 in H. In this paper, we establish a structural characterization of $$G \times C_{2n+1}$$ G×C2n+1 . Further, we discuss Eulerian, Hamiltonian and planar properties of $$G \times C_{2n+1}$$ G×C2n+1 .

We study the steady, two-dimensional boundary layer flow and heat transport caused by a stretching cylinder immersed in an incompressible, viscous nanoliquid. The thermal analysis has been done for prescribed surface temperature as well as for prescribed heat flux. The governing partial differential equations in cylindrical form for momentum and heat transfer are reduced to non-linear ordinary differential equations by using similarity transformation. The differential transform method is used to solve these non-linear ordinary differential equations, under appropriate boundary conditions, in the form of power series. The inverse Domb–Sykes plots between number of terms against ratio of consecutive coefficients of the power series are sketched to determine the minimum number of terms required in the series to ensure convergence. A rational approximation in the form of Pad$$\acute{e}$$ e´ approximant is then applied to increase the convergence of the power series. Results for skin friction coefficient and surface heat transfer have been presented as a property of nanoparticle volume fraction, curvature parameter and Prandtl number. Dilute concentration $$(0<\phi \le 0.2)$$ (0<ϕ≤0.2) of copper-nanoparticle is considered. Comparison of numerical results are made with previously published works in some limiting cases. The problem is shown to reduce to the stretching sheet case in the absence of transverse curvature.

AbstractFormally, a signed graph (sigraph) Sis a pair (G, σ) that consists of a graph G= (V, E) and a sign mapping or signature σfrom Eto the sign group {+, –}. Given a sigraph Sand a positive integer t, the t-path sigraph(S)tof Sis a sigraph whose vertex set is V(S) and two vertices are adjacent if and only if there exist a path of length tbetween these vertices and then by defining its sign st(e) to be ‘-’ if and only if in every such path of length tin Sall the edges are negative. The negation η(S) of a sigraph Sis a sigraph obtained from Sby reversing the sign of every edge of S. Two sigraphs S1and S2on the same underlying graph are switching equivalent if it is possible to assign signs ‘+’ (‘plus’) or ‘-’ (‘minus’) to the vertices of S1such that by reversing the sign of each of its edges that has received opposite signs at its ends one obtains S2. In this paper, we characterize sigraphs whose negations are switching equivalent to their t-path sigraphs for t= 3.

A signed graphis a graph in which every edge is designated to be either positiveor negative; it is balancedif every cycle contains an even number of negative edges. A marked signed graphis a signed graph each vertex of which is designated to be positive or negative, and it is consistentif every cycle in the signed graph possesses an even number of negative vertices. Signed line graphL(S) of a given signed graph $$S=(G, \sigma )$$ S=(G,σ) , as given by Behzad and Chartrand (Element Math 24(3):49–52, 1969), is the signed graph with the standard line graph L(G) of Gas its underlying graph and whose edges are assigned the signs according to the rule: for any $$e_ie_j \in {E(L(S))}$$ eiej∈E(L(S)) , $$e_ie_j \in {E^-{L(S)}}\Leftrightarrow $$ eiej∈E-L(S)⇔ the edges $$e_i$$ ei and $$e_j$$ ej of Sare both negative in S. Iterated signed line graphs $$L^k(S)$$ Lk(S) =$$L(L^{k-1}(S))$$ L(Lk-1(S)) , $$k\in N$$ k∈N , S:= $$L^0(S)$$ L0(S) is defined similarly. Further, L(S) is S-consistentif to each vertex eof L(S), which is an edge of S, one assigns the sign $$\sigma (e)$$ σ(e) then the resulting marked signed graph $$(L(S))_\mu $$ (L(S))μ is consistent. In this paper, we give a characterization of signed graphs Swhose iterated signed line graphs $$L^k(S)$$ Lk(S) are balanced or S-consistent.

A signed graph(sigraph) is an ordered pair $$S = (S^u, \sigma ),$$ S=(Su,σ), where $$S^u$$ Su is a graph $$G = (V, E)$$ G=(V,E) and $$\sigma :E\rightarrow \{+,-\}$$ σ:E→{+,-} is a function from the edge setEof $$S^u$$ Su into the set {+, −}. The canonical markingonSis defined as: for each vertex $$v \in V(S),$$ v∈V(S), $$\displaystyle \mu _\sigma (v) = \prod _{e_j \in E_{v}} \sigma (e_j),$$ μσ(v)=∏ej∈Evσ(ej), where $$E_{v}$$ Ev is the set of edges $$e_j$$ ej incident atvinS. A vertexvis called negativeif the value of marking of vis negative. Let Sbe canonically marked, then a cycle Zin Sis said to be canonically consistentif it contains an even number of negative vertices. If every cycle in Sis canonically consistent, then Sis called canonically consistent. In this paper, we characterize canonically consistent semi-total line sigraphs.

Let Rbe a commutative ring with unity. The co-maximal graph Γ(R)is the graph with vertex set Rand two vertices aand bare adjacent if Ra+Rb=R. In this paper, we characterize rings for which the co-maximal graph Γ(R)is a line graph of some graph Gand determine rings for which Γ(R)is Eulerian or Hamiltonian. We also find the diameter and girth of Γ(R).

AbstractLet be a commutative ring with unity. The co-maximal graph is the graph with vertex set and two vertices and are adjacent if . In this paper, we characterize rings for which the co-maximal graph is a line graph of some graph and determine rings for which is Eulerian or Hamiltonian. We also find the diameter and girth of .

AbstractA signed graph(or sigraphin short) Sis a graph Gwhere the edges are weighted either + 1 or − 1 called the signsof the edges. Considering Zn′, the integers modulo n, where nis a positive integer greater than 1, special kind of structure called unitary addition Cayley graphdenoted by Gnexists in the literature. On the unitary addition Cayley graph Gn, we define four different unitary addition Cayley sigraph denoted by , , and , by assigning the signs to the edges of Gnby a certain rule. The study in this paper concentrates on the discussion of one such signed structure called unitary addition Cayley ring sigraph, for different properties such as balancing, clusterability, canonically consistency and sign-compatibility.

AbstractA signed graph(or sigraphin short) Sis a graph Gwhere the edges are weighted either+1 or – 1 called the signsof the edges. For a given sigraph, a special kind of structure called common-edge sigraphand second iterated line sigraphexists in literature which has been shown to be isomorphic to each other when the considered sigraph is only positively signed. Particularly, this paper concentrates on algorithmically characterizing sigraphs whose common-edge sigraphs and second iterated line sigraphs remains invariant under switching due to Abelsonand Rosenbergand an algorithm to obtain commmon-edge sigraph and second iterated line sigraph in O(n3) steps.

A $$\text{signed\,graph}$$ signed\,graph (or $$\text{sigraph}$$ sigraph in short) $$S$$ S is a graph $$G$$ G in which each edge $$x$$ x carries a value $$s(x) \in \{+1, -1\}$$ s(x)∈{+1,-1} called its $$sign$$ sign denoted specially as $$S = (G, s)$$ S=(G,s) . A sigraph $$S$$ S is $$\text{sign-compatible}$$ sign-compatible if there exists a marking $$\mu $$ μ of its vertices such that the end vertices of every negative edge receive ‘-’ signs in $$\mu $$ μ and no positive edge in $$S$$ S has both of its ends assigned ‘-’ sign in $$\mu $$ μ . In this paper, we write algorithms to detect sign-compatibility of a given sigraph and obtain optimal algorithm with complexity $$O(n^2)$$ O(n2) .

In this paper, we consider a nonlinear multiobjective programming problem where functions involved are non-differentiable. The use of directional derivative in association with a hypothesis of an invex kind on the set has been of much interest in the recent past. Recently under preinvexity assumptions on fi′(xo;ηi(x,xo)) and g′j(xo;θj(x,xo)), necessary and sufficient optimality conditions for a non-smooth multiobjective optimisation problem under generalised class of dI-V-type I functions have been derived. Working in this direction, we introduce a new class of (φ, dI)-V-type I functions and illustrate through various non-trivial examples that this class is non-empty and extends some known classes introduced in the literature. Also, we obtain sufficient optimality conditions to enable a feasible solution of the primal problem to be its weak efficient/efficient solution. Then through an example, we illustrate the relevance of the sufficient optimality theorem obtained to get the efficient solution of the problem. Further, we formulate Wolfe type and Mond-Weir type multiobjective dual programs and establish various duality theorems under this newly introduced class of functions.

In this paper, we consider a multiobjective optimisation problem where the objective and constraint functions involved are non-differentiable. We introduce a new class of functions namely ( d I − ρ − σ )-V-type I functions and illustrate through non-trivial examples that the class introduced is non-empty. We then obtain sufficient optimality conditions under the newly introduced class of functions and derive various weak, strong, converse and strict converse duality results for Wolfe type and Mond-Weir type dual programmes in order to relate the efficient and weak efficient solutions of primal and dual problems.

AbstractA signed graph(or sigraphin short) is an ordered pair S= (Su, σ), where Suis a graph G= (V, E) and σ: E→ {+, -} is a function from the edge set Eof Suinto the set {+, -}. A sigraph Sis called signed-regularif the number of positive edges, d+(v) incident at a vertex vin S, is independent of the choice of vin Sand the number of negative edges, d-(v) incident at a vertex vin Sis also independent of the choice of vin S. In this paper, we characterize the signed-regularity of line sigraph, x-line sigraph and total sigraph.

AbstractA signed graph(or, sigraphin short) is a graph Gin which each edge xcarries a value s(x) ∊ { -1, +1} called its sign. Given a sigraph S, a new sigraph CE(S), called the common-edge sigraphof S, is that sigraph whose vertex-set is the set of pairs of adjacent edges in Sand two vertices of CE(S) are adjacent if the pairs of adjacent edges of Shave exactly one edge in common, with the same sign as that of the common edge. In this paper, we study common-edge sigraphs and characterize sigraphs whose common-edge sigraphs and second iterated line sigraphs are switching equivalent. Also, we determine the sigraphs Sfor which their common-edge sigraphs are balanced, S-consistent and S-cycle-compatible, respectively.

This paper presents the results of M60 grade of concrete. M60 grade of concrete is achieved by maximum density technique. Concrete is brittle and weak in tension and develops cracks during curing and due to thermal expansion / contraction over a period ot time. Thus the effect of addition of 1% steel fibre is studied. For ages, concrete has been one of the widely used materials for construction. When cement is manufactured, every one ton of cement produces around one ton of carbon dioxide leading to global warming and also as natural resources are finishing, so use of supplementary cementitious material like alccofine and flyash is used as partial replacement of cement is considered. The effect of binary and ternary blend on the strength characteristics is studied. The results indicate that the concrete made with alccofine and flyash generally show excellent fresh and hardened properties. The ternary system that is Portland cement-fly ash-Alccofine concrete was found to increase the strength of concrete when compared to concrete made with Portland cement or even from Portland cement and fly ash.