A matrix with entries [Formula: see text] is graphical if it is symmetric and all its diagonal entries are zero. Let [Formula: see text], [Formula: see text] and [Formula: see text] be graphs defined on the same set of vertices. The graph [Formula: see text] is said to be the matrix product of graphs [Formula: see text] and [Formula: see text], if [Formula: see text], where [Formula: see text] is the adjacency matrix of the graph [Formula: see text]. In such a case, we say that [Formula: see text] and [Formula: see text] are companions of each other. The main purpose of this paper is to design an algorithm to check whether a given graph [Formula: see text] has a companion. We derive conditions on [Formula: see text] and [Formula: see text] so that the generalized wheel graph, denoted by [Formula: see text], has a companion and also show that the [Formula: see text]th power of the path graph [Formula: see text] has no companion. Finally, we indicate a possible application of the algorithm in a problem of coloring of edges of the complete graph [Formula: see text].

Chain graphs and threshold graphs are special classes of graphs which have maximum spectral radius among bipartite graphs and connected graphs with given order and size, respectively. In this article, we focus on some of linear algebraic tools like rank, determinant, and permanent related to the adjacency matrix of these types of graphs. We derive results relating the rank and number of edges. We also characterize chain/threshold graphs with nonzero determinant and permanent.

Las variaciones en el curso de la arteria maxilar se describen a menudo, con sus relaciones con el músculo pterigoideo lateral. En el presente caso informamos una variación exclusiva en el curso de la arteria maxilar que no fue publicada antes. En un cadáver masculino de 75 años arteria maxilar derecho estaba pasando por el bucle del nervio auriculo-temporal. La arteria meníngea media provenía de la arteria maxilar con un bucle del nervio auriculo-temporal. La arteria maxilar pasaba profunda con respecto al nervio dentario inferior pero superficial al nervio lingual. El conocimiento de estas variaciones es importante para el cirujano y también serviría para explicar la posible participación de estas variaciones en la etiología del dolor mandibular. Variations in the course of the maxillary artery are often described with its relations to the lateral pterygoid muscle. In the present case we report a unique variation in the course of the maxillary artery which was not reported before. In a 75 years old male cadaver the right maxillary artery passed through the loop of the auriculotemporal nerve. The middle meningeal artery was arising from the maxillary artery within the nerve loop of auriculotemporal nerve. Further the maxillary artery passed deep to the inferior alveolar nerve but superficial to the lingual nerve. The knowledge of these variations is important for surgeons and it would also explain the possible involvement of these variations in etiology of the craniomandibular pain.

E. Sampath Kumar and L. Pushpalatha [4] introduced a generalized version of complement of a graph with respect to a given partition of its vertex set. Let G = (V,E) be a graph and P = {V₁, V₂,...,Vk} be a partition of V of order k ≥ 1. The k-complement GPk of G with respect to P is defined as follows: For all Vi and Vj in P, i ≠ j, remove the edges between Vi and Vj , and add the edges which are not in G. Analogues to self complementary graphs, a graph G is k-self complementary (k-s.c.) if GPk ≅ G and is k-co-self complementary (k-co.s.c.) if GPk ≅ Ġ with respect to a partition P of V (G). The mth power of an undirected graph G, denoted by Gm is another graph that has the same set of vertices as that of G, but in which two vertices are adjacent when their distance in G is at most m. In this article, we study powers of cycle graphs which are k-self complementary and k-co-self complementary with respect to a partition P of its vertex set and derive some interesting results. Also, we characterize k-self complementary C2n and the respective partition P of V (C2n). Finally, we prove that none of the C2n is k-co-self complementary for any partition P of V (C2n).

In this paper, we introduce the notion of perfect matching property for a k-partition of vertex set of given graph. We consider nontrivial graphs G and GPk , the k-complement of graph G with respect to a kpartition of V(G), to prove that A(G)A(GPk ) is realizable as a graph if and only if P satis_es perfect matching property. For A(G)A(GPk ) = A(Γ) for some graph Γ, we obtain graph parameters such as chromatic number, domination number etc., for those graphs and characterization of P is given for which GPk and Γ are isomorphic. Given a 1-factor graph G with 2n vertices, we propose a partition P for which GPk is a graph of rank r and A(G)A(GPk ) is graphical, where n ≤ r ≤ 2n. Motivated by the result of characterizing decomposable Kn,n into commuting perfect matchings [2], we characterize complete k-partite graph Kn1,n2,...,nk which has a commuting decomposition into a perfect matching and its k-complement.

A bipartite graph G can be treated as a (1,1) bipartite graph in the sense that, no two vertices in the same part are at distance one from each other. A (2,2) bipartite graph is an extension of the above concept in which no two vertices in the same part are at distance two from each other. In this article, analogous to complete (1,1) bipartite graphs which have the maximum number of pairs of vertices having distance one between them, a complete (2,2) bipartite graph is defined as follows. A complete (2,2) bipartite graph is a graph which is (2,2) bipartite and has the maximum number of pairs of vertices (u,v) such that d(u,v)=2. Such graphs are characterized and their properties are studied. The expressions are derived for the determinant, the permanent and spectral properties of some classes of complete (2,2) bipartite graphs. A class of graphs among complete (2,2) bipartite graphs having golden ratio in their spectrum is obtained.