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All graphs under consideration are finite, simple, connected, and undirected. Adjacency matrix of a graph G is 0,1 matrix A = a i j = 0 , i f v i = v j o r d v i , v j ≥ 2 1 , i f d v i , v j = 1. . Here in this paper, we discussed new type of adjacency matrix known by 1-2 adjacency matrix defined as A 1,2 G = a i j = 0 , i f v i = v j o r d v i , v j ≥ 3 1 , i f d v i , v j = 2 , from eigenvalues of the graph, we mean eigenvalues of the 1-2 adjacency matrix. Let T n c be the set of the complement of trees of order n . In this paper, we characterized a unique graph whose least eigenvalue is minimal among all the graphs in T n c .

In this paper, according to the known results of some normalized permutation polynomials with degree 5 over F 2 n , we determine sufficient and necessary conditions on the coefficients b 1 , b 2 ∈ F 2 n 2 such that f x = x 3 x ¯ 2 + b 1 x 2 x ¯ + b 2 x permutes F 2 n . Meanwhile, we obtain a class of complete permutation binomials over F 2 n .

An m - Ł ℘ , ℘ − 1 , … , 1 labeling of a simple graph G is a mapping τ from the vertex set ∨ G to 0,1 , … , m such that τ u − τ v ≥ ℘ + 1 − d u , v , ∀ u , v ∈ ∨ G , where length of the shortest route connecting u and v is represented by d u , v . The smallest m for which there exists a m - Ł ℘ , ℘ − 1 , … , 1 labeling of G is known as the Ł ℘ , ℘ − 1 , … , 1 labeling number of G , and it is described by λ ℘ G . We define m - Ł ′ ℘ , ℘ − 1 , … , 1 as the same as the m - Ł ℘ , ℘ − 1 , … , 1 labeling if τ is one to one. The Ł ′ ℘ , ℘ − 1 , … , 1 labeling number of G represented by λ ℘ , ℘ − 1 , … , 1 ′ G and is called minimum span. In this paper, we prove that the circulant graphs with specified generating sets admit m - Ł ℘ , ℘ − 1 , … , 1 and m - Ł ′ ℘ , ℘ − 1 , … , 1 labeling and also find λ ℘ G and λ ℘ , ℘ − 1 , … , 1 ′ G . Moreover, we find the Ł ℘ , ℘ − 1 , … , 1 labeling number of any simple graph with diameter less than ℘ .

Let G be a graph with n vertices, and let L G and Q G denote the Laplacian matrix and signless Laplacian matrix, respectively. The Laplacian (respectively, signless Laplacian) permanental polynomial of G is defined as the permanent of the characteristic matrix of L G (respectively, Q G ). In this paper, we show that almost complete graphs are determined by their (signless) Laplacian permanental polynomials.

Let G be a simple, connected, and finite graph. For every vertex v ∈ V G , we denote by N G v the set of neighbours of v in G . The locating-dominating number of a graph G is defined as the minimum cardinality of W ⊆ V G such that every two distinct vertices u , v ∈ V G \ W satisfies ∅ ≠ N G u ∩ W ≠ N G v ∩ W ≠ ∅ . A graph G is called k -regular graph if every vertex of G is adjacent to k other vertices of G . In this paper, we determine the locating-dominating number of k -regular graph of order n , where k = n − 2 or k = n − 3 .

For the study and valuation of social graphs, which affect an extensive range of applications such as community decision-making support and recommender systems, it is highly recommended to sustain the resistance of a social graph G to active attacks. In this regard, a novel privacy measure, called the k , l -anonymity, is used since the last few years on the base of k -metric antidimension of G in which l is the maximum number of attacker nodes defining the k -metric antidimension of G for the smallest positive integer k . The k -metric antidimension of G is the smallest number of attacker nodes less than or equal to l such that other k nodes in G cannot be uniquely identified by the attacker nodes. In this paper, we consider four families of wheel-related social graphs, namely, Jahangir graphs, helm graphs, flower graphs, and sunflower graphs. By determining their k -metric antidimension, we prove that each social graph of these families is the maximum degree metric antidimensional, where the degree of a vertex is the number of vertices linked with that vertex.

In this paper, we focus on the degree of the greatest common divisor ( gcd ) of random polynomials over F q . Here, F q is the finite field with q elements. Firstly, we compute the probability distribution of the degree of the gcd of random and monic polynomials with fixed degree over F q . Then, we consider the waiting time of the sequence of the degree of gcd functions. We compute its probability distribution, expectation, and variance. Finally, by considering the degree of a certain type gcd , we investigate the probability distribution of the number of rational (i.e., in F q ) roots (counted with multiplicity) of random and monic polynomials with fixed degree over F q .