We introduce the category of optiongraphs and option-preserving maps as a model to study impartial combinatorial games.Outcomes, remoteness, and extended nim-values are preserved under option-preserving maps.We show that the four isomorphism theorems from universal algebra are valid in this category.Quotient optiongraphs, including the minimum quotient, provide simplifications that can help in the analysis of games.

The connectivity matrix of a graph is the matrix whose off-diagonal i-j entry is the maximum number of internally vertex disjoint paths between the vertices i and j with only zeros on its diagonal.In this article, lower and upper estimates for the largest eigenvalue of such matrices are investigated.Bounds in terms of the underlying graph's average degree are provided and it is shown that they can be utilized to refine previous bounds on the energy of connectivity matrices.

Consider a family F of k-subsets of an ambient (k 2 -k + 1)-set such that no pair of k-subsets in F intersects in exactly one element.In this article, it is shown that the maximum size offor every k > 1.

In 1968, Vizing conjectured that for every pair of graphs X and Y, the inequality (X Y ) (X)(Y ) holds, where stands for the domination number and X Y is the Cartesian product of X and Y.In a breakthrough result, Clark and Suen [Electron.J. Combin.7 (2000) #N4] proved that (X Y ) 1 2 (X)(Y ).In this paper, a lower bound for (X Y Z) is obtained using projections in the space.It is shown how the obtained bound implies the mentioned result of Clark and Suen.

The cozero divisor graph Γ (R) of a commutative ring R is a simple graph whose vertex set is the set of non-zero non-unit elements of R such that two distinct vertices x and y of Γ (R) are adjacent if and only if x / ∈ Ry and y / ∈ Rx, where Rx is the ideal generated by x.In this article, the independent domination polynomial of Γ (Zn) is found for n ∈ {p1p2, p1p2p3, p n 1 1 p2}, where pi's are primes, n1 is an integer greater than 1, and Zn is the integer modulo ring.It is shown that the independent domination polynomial of Γ (Zp 1 p 2 ) has only one real root.It is also proved that these polynomials are not unimodal but are log-concave under certain conditions.

Let G be a graph with no multiple edges and loops.A subset S of the vertex set of G is a dominating set ofhas at least k neighbors in S and the subgraph G[S] is connected.The domination number of G is the number of vertices in a minimum dominating set of G, denoted by (G).The connected k-domination number of G, denoted by c k (G), is the minimum cardinality of a connected k-dominating set of G.For k = 1, we simply write c(G).It is known that the bounds c 2 (G) (G) + 1 and c 2 (G) c(G) + 1 are sharp.In this research article, we present the necessary condition of the connected graphs G with c 2 (G) = (G) + 1 and the necessary condition of the connected graphs G with c 2 (G) = c(G) + 1.Moreover, we present a graph construction that takes in any connected graph with r vertices and gives a graph G with c 2 (G) = r, c(G) = r -1, and (G) {r -1, r -2}.

A classical theorem independently due to Gallai and Roy states that a graph G has a proper k-coloring if and only if G has an orientation without coherent paths of length k. An analogue of this result for signed graphs is proved in this article.