Based on a motivation coming from the study of the metric structure of the category of finite dimensional vector spaces over a finite field $$\mathbb {F}$$ F , we examine a family of graphs, defined for each pair of integers $$1 \le k \le n$$ 1 ≤ k ≤ n , with vertex set formed by all injective linear transformations $$\mathbb {F}^k \rightarrow \mathbb {F}^n$$ F k ? F n and edges corresponding to pairs of mappings, $$f$$ f and $$g$$ g , with $$\lambda (f,g)= \dim \mathrm{Im }(f-g)=1 $$ ? ( f , g ) = dim Im ( f - g ) = 1 . For $$\mathbb {F}\cong \mathrm{GF }(q)$$ F ? GF ( q ) , this graph will be denoted by $$\mathrm{INJ }_q(k,n)$$ INJ q ( k , n ) . We show that all such graphs are vertex transitive and Hamiltonian and describe the full automorphism group of each $$\mathrm{INJ }_q (k,n)$$ INJ q ( k , n ) for $$k<n$$ k < n . Using the properties of line-transitive groups, we completely determine which of the graphs $$\mathrm{INJ }_q (k,n)$$ INJ q ( k , n ) are Cayley and which are not. The Cayley ones consist of three infinite families, corresponding to pairs $$(1,n),\,(n-1,n)$$ ( 1 , n ) , ( n - 1 , n ) , and $$(n,n)$$ ( n , n ) , with $$n$$ n and $$q$$ q arbitrary, and of two sporadic examples $$\mathrm{INJ }_{2} (2,5)$$ INJ 2 ( 2 , 5 ) and $$\mathrm{INJ }_{2}(3,5)$$ INJ 2 ( 3 , 5 ) . Hence, the overwhelming majority of our graphs is not Cayley.