Let G be a signed graph, where G = (V; E) is the underlying simple graph and  : E(G) → {± 1} is the sign function on E(G). In this paper, we obtain k-th signed spectral moments and k-th signed Laplacian spectral moments of a signed graph G , together with coe cients of their signed characteristic polynomial and signed Laplacian characteristic polynomial are calculated.

For a given graph G, its P-energy is the sum of the absolute values of the eigenvalues of the P-matrix of G. In this article, we explore the P-energy of Generalized Petersen graphs G(p; k) for various vertex partitions such as independent, domatic, total domatic and k-ply domatic partitions and partition containing a perfect matching in G(p; k). Further, we present a python program to obtain the P-energy of G(p; k) for the vertex partitions under consideration and examine the relation between them.

In a graph G=(V, E), an independent set I(G) is a subset of the vertices of G such that no two vertices in I(G) are adjacent. Any maximum independent set of a graph is called a diagonal of the graph. Let c be a proper (r+1)-coloring of an r-regular graph G. A vertex v in G is said to be rainbow with respect to c if every color appears in the closed neighborhood N[v]=N(v)∪ {v}. Given a diagonal I of G, the coloring c is said to be silver with respect to I if every v∈ I is rainbow with respect to c. G is called silver if it admits a silver coloring with respect to some diagonal. In [1], the authors introduced silver coloring and the following question is raised “ Find classes of r-regular graphs G, that G is a silver graph". This paper is aimed toward study this question for the Generalized Petersen graphs. In this paper we show that, if n≡ 0 (mod4) and k is odd, then P(n, k) is a totally silver graph. Also, for every natural number n, the existence of silver coloring for Generalized Petersen graphs P(n, 1), P(n, 2) except for n=5, this is well-known Petersen graph, P(n, 3) except for n=10, 14 and 26. Also, for any k>2, P(2k+1, k), and for any k>3, P(3k+1, k), and for any k>3, k ≠ 5, 9 P(3k-1, k) are silver graphs.

A signed graph is called a parity signed graph if there exists a bijective mapping such that for each edge in , and have same parity if , and opposite parity if . The \emph{rna} number of is the least number of negative edges among all possible parity signed graphs over . Equivalently, is the least size of an edge-cut of that has nearly equal sides. In this paper, we show that for the Generalized Petersen graph , lies between and . Moreover, we determine the exact value of for . The \emph{rna} numbers of some famous Generalized Petersen graphs, namely, Petersen graph, D\" urer graph, M\" obius-Kantor graph, Dodecahedron, Desargues graph and Nauru graph are also computed. Recently, Acharya, Kureethara and Zaslavsky characterized the structure of those graphs whose \emph{rna} number is . We use this characterization to show that the smallest order of a -regular graph having \emph{rna} number is . We also prove the smallest order of -regular graphs having \emph{rna} number is bounded above by . In particular, we show that the smallest order of a cubic graph having \emph{rna} number is 10.

Cop Robber game is a two player game played on an undirected graph. In this game, the cops try to capture a robber moving on the vertices of the graph. The cop number of a graph is the least number of cops needed to guarantee that the robber will be caught. We study cop-edge critical graphs, i. e. graphs G such that for any edge e in E(G) either c(G-e) < c(G) or c(G-e) > c(G). In this article, we study the edge criticality of Generalized Petersen graphs and Paley graphs.

The aim of this paper is to determine an upper bound for the number of non-co-spectral permutation graphs in terms of automorphism group of a graph G. As a corollary, we determine the eigenvalues of all permutation graphs P (Cn), where  2 Aut(Cn).

In this paper, we introduce a suitable generalization of Cayley graphs that is de ned over polygroups (GCP-graph) and give some examples and properties. Then, we mention a generalization of NEPS that contains some known graph operations and apply to GCP-graphs. Finally, we prove that the product of GCP-graphs is again a GCP-graph.

Let G = (V, E) be a graph. A double Roman dominating function (DRDF) of G is a function f: V →, { 0,1,2,3} such that, for each v ,V with f(v) = 0, there is a vertex u adjacent to v with f(u) = 3 or there are vertices x and y adjacent to v such that f(x) = f(y) = 2 and for each v ,V with f(v) = 1, there is a vertex u adjacent to v with f(u) > 1. The weight of a DRDF f is f(V ) =∑, v, V f(v). Let n and k be integers such that 3 ≥,2k + 1 ≥,n. The Generalized Petersen graph GP(n,k) = (V, E) is the graph with V = {u1,u2,…, , un},{v1,v2,…, , vn} and E = {uiui+1,uivi,vivi+k: 1 ≥,i ≥,n}, where addition is taken modulo n. In this paper, we _rstly prove that the decision problem associated with double Roman domination is NP-complete even restricted to planar bipartite graphs with maximum degree at most 4. Next, we give a dynamic programming algorithm for computing a minimum DRDF (i. e., a DRDF with minimum weight along all DRDFs) of GP(n,k) in O(n81k) time and space and so a minimum DRDF of GP(n, O(1)) can be computed in O(n) time and space.

Let G be a simple connected graph. The Generalized polarity Wiener index of G is defined as the number of unordered pairs of vertices of G whose distance is k. Some formulas are obtained for computing the Generalized polarity Wiener index of the Cartesian product and the tensor product of graphs in this article.