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Let R be the real field and M m , n (R) be the set of all m × n matrices over R , where m , n ≥ 2. For a square matrix A ∈ M m , m (R) , τ (A) denotes the trace of A (the sum of all diagonal entries of A). The symmetry trace graph Γ t (M m , n (R)) of M m , n (R) is defined to be a graph with vertex set of all nonzero matrices in M m , n (R) and two vertices A and B are adjacent if and only if τ (A B ′) = 0 , where B ′ is the transpose of B. Clearly, Γ t (M m , n (R)) is undirected and without loops. In the present paper, by studying maximum cliques of Γ t (M m , n (R)) , we determine the form of an arbitrary automorphism of Γ t (M m , n (R)). [ABSTRACT FROM AUTHOR]

Let T be a tree with vertex set V (T) = { v 1 , v 2 , ... , v n }. The adjacency matrix A (T) of T is an n × n matrix (a i j) , where a i j = a j i = 1 if v i is adjacent to v j and a i j = 0 if otherwise. In this paper, we consider the multiplicity of − 1 as an eigenvalue of A (T) , which is written as m (T , − 1). It is proved that among all trees T with p ≥ 2 pendant vertices, the maximum value of m (T , − 1) is p−1, and for a tree T with p ≥ 2 pendant vertices, m (T , − 1) = p − 1 if and only if T = P n with n ≡ 2 (m o d 3) , or T is a tree in which d (v , u) ≡ 2 (m o d 3) for any pendant vertex v and any major vertex u of T, where a major vertex is a vertex of degree at least 3 and d (v , u) is the distance between v and u. [ABSTRACT FROM AUTHOR]

Let R be a ring with identity. The intersection graph of ideals of R, written as I (R) , is a graph which has all non-trivial left ideals of R as its vertices and two distinct non-trivial left ideals I 1 , I 2 of R are adjacent if and only if I 1 ∩ I 2 ≠ 0. Let F q be a finite field with q elements, M n (F q) the ring of all n × n matrices over F q . In this article, all automorphisms of I (M n (F q)) are determined definitely. [ABSTRACT FROM AUTHOR]

Let F q be a finite field with q elements, n ≥ 2 a positive integer, T (n , q) the semigroup of all n × n upper triangular matrices over F q under matrix multiplication, T ∗ (n , q) the group of all invertible matrices in T (n , q) , P T ∗ (n , q) the quotient group of T ∗ (n , q) by its center. The one-divisor graph of T (n , q) , written as n (q) , is defined to be a directed graph with T (n , q) as vertex set, and there is a directed edge from A ∈ T (n , q) to B ∈ T (n , q) if and only if r (A B) = 1 , i.e. A and B are, respectively, a left divisor and a right divisor of a rank one matrix in T (n , q). The definition of n (q) is motivated by the definition of zero-divisor graph of T (n , q) , which has vertex set of all nonzero zero-divisors in T (n , q) and there is a directed edge from a vertex A to a vertex B if and only if A B = 0 , i.e. r (A B) = 0. The automorphism group of zero-divisor graph of T (n , q) was recently determined by Wang [A note on automorphisms of the zero-divisor graph of upper triangular matrices, Lin. Alg. Appl.465 (2015) 214–220.]. In this paper, we characterize the automorphism group of one-divisor graph n (q) of T (n , q) , proving that Aut ( n (q)) ≃ P T ∗ (n , q) × Aut (F q) × H , where Aut (F q) is the automorphism group of field F q , H is a direct product of some symmetric groups. Besides, an application of automorphisms of n (q) is given in this paper. [ABSTRACT FROM AUTHOR]

Let R be a ring and Z (R) = Z l (R) ∪ Z r (R) , where Z l (R) and Z r (R) are the sets of all left and right zero-divisors of R, respectively. The zero-divisor graph of R, denoted by Γ (R) , is a simple undirected graph with vertex set Z ∗ (R) = Z (R) ∖ { 0 } , and two distinct vertices a , b ∈ Z ∗ (R) are adjacent if and only if ab = 0 or ba = 0. Let n ≥ 2 , Z n the ring of integers modulo n, and M a t n (q) the ring of all n × n matrices over a finite field of q elements. In this article, using the technique on characteristic matrices, we give the value of fixing number of Γ (∏ i = 1 n Z 2). Moreover, we calculate the fixing number and metric dimension of Γ (Z n) and Γ (M a t n (q)). [ABSTRACT FROM AUTHOR]

The orthogonality graph O (R) of a ring R is the undirected graph with vertex set consisting of all nonzero two-sided zero divisors of R , in which for two vertices x and y (needless distinct), x ∼ y is an edge if and only if x y = y x = 0. Let n ≥ 2 , Mat n (F) be the set of all n × n matrices over a finite field F , and R n (F) the subset of Mat n (F) consisting of all rank one upper triangular matrices. In this paper, we describe the full automorphism group, and using the technique of generalized equivalent canonical form of matrices, we compute the fixing number of O (R n (F)) , the induced subgraph of O ( Mat n (F)) with vertex set R n (F). [ABSTRACT FROM AUTHOR]

An undirected graph G is determined by its T-gain spectrum (DTS) if every T-gain graph cospectral to G is switching equivalent to G. We show that the complete graph Kn and the graph Kn−e obtained by deleting an edge from Kn are DTS, the star K1,n is DTS if and only if n ≤ 2, and an odd path P2m+1 is not DTS if m ≥ 2. We give an operation for constructing cospectral T-gain graphs and apply it to show that a tree of arbitrary order (at least 5) is not DTS. [ABSTRACT FROM AUTHOR]

The inclusion graph of a finite group G , written as ℐ n (G) , is defined to be an undirected graph whose vertices are all nontrivial subgroups of G , and two distinct vertices H , K are adjacent if and only if either H ⊂ K or K ⊂ H. For a graph Γ with vertex set V (Γ) , a set of vertices ℱ ⊆ V (Γ) is called a fixing set of Γ if the only automorphism of Γ that fixes every element in ℱ is the identity. The fixing number of Γ is the smallest size of a fixing set of Γ. In this paper, we determine the finite nilpotent groups whose inclusion graphs are planar. Moreover, using the technique of characteristic matrices, we characterize the fixing sets and give the exact value on the fixing number of the inclusion graphs for finite cyclic groups. [ABSTRACT FROM AUTHOR]

Let F be a finite field, R = T n (F) the ring of all n × n upper triangular matrices over F , Z (R) the set of all zero-divisors of R , i.e. Z (R) consists of all n × n upper triangular singular matrices over F. The total graph of R = T n (F) , denoted by 𝒯 (Γ (R)) , is a graph with all elements of R as vertices, and two distinct vertices x , y ∈ R are adjacent if and only if x + y ∈ Z (R). In this paper, we determine all automorphisms of the total graph 𝒯 (Γ (R)) of R = T n (F). [ABSTRACT FROM AUTHOR]